New England Dynamics and Number Theory Seminar

Spring 2026

Diophantine approximation for hypersurfaces
Alexander Smith, Northwestern University, 29 January 2026
Abstract: Among the nondegenerate C⁴ hypersurfaces, we characterize the rational quadrics as the hypersurfaces that are the least well approximated by rational points. For all other hypersurfaces, we give a heuristically sharp lower bound for the number of rational points near them, improving the sensitivity of prior results of Beresnevich and Huang. Our methods are dynamical, involving the application of Ratner’s theorems for unipotent orbits, and we will show how our work relates to the dynamical resolution of the Oppenheim conjecture by Margulis.
Video

Asymptotically large free semigroups in Zariski dense discrete subgroups of Lie groups
Alexander Skenderi, UW Madison, 5 February 2026
Abstract: An important quantity in the study of discrete groups of isometries of Riemannian manifolds, Gromov hyperbolic spaces, and other interesting geometric objects is the critical exponent. For a discrete subgroup of isometries of the quaternionic hyperbolic space or octonionic projective plane, Kevin Corlette established in 1990 that the critical exponent detects whether a discrete subgroup is a lattice or has infinite covolume. Precisely, either the critical exponent equals the volume entropy, in which case the discrete subgroup is a lattice, or the critical exponent is less than the volume entropy by some definite amount, in which case the discrete subgroup has infinite covolume. In 2003, Leuzinger extended this gap theorem for the critical exponent to any discrete subgroup of a Lie group having Kazhdan’s property (T) (for instance, a discrete subgroup of SL(n,R), where n is at least 3). In this talk, I will present a result which shows that no such gap phenomenon holds for discrete semigroups of Lie groups. More precisely, for any Zariski dense discrete subgroup of a Lie group, there exist free, finitely generated, Zariski dense subsemigroups whose critical exponents are arbitrarily close to that of the ambient discrete subgroup. As an application, we show that the critical exponent is lower semicontinuous in the Chabauty topology whenever the Chabauty limit of a sequence of Zariski dense discrete subgroups is itself a Zariski dense discrete subgroup.
Video

Unique Equilibrium States for Viana Maps for Small Potentials
Kecheng Li, Tufts University, 26 February 2026
Abstract: We study the thermodynamic formalism for Viana maps (skew products that couple an expanding circle map with a small perturbation of a quadratic map on the fibers). Working within the Climenhaga–-Thompson framework, we show that for every Hölder potential whose oscillation is below an explicit threshold, there is a unique equilibrium state. The main step is a uniform control of recurrence to the critical region in the fibers, where the derivative degenerates. This yields the pressure gap and the specification estimates needed to apply the method and removes the principal obstruction. These conclusions are robust under sufficiently small perturbations of the reference map.
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Accumulation points of normalized approximations
Kavita Dhanda, University of Houston, 5 March 2026
Abstract: Consider the collection of all accumulation points of normalized integer vector translates of points qα with α ∈ R^d and q ∈ Z. For each normalization factor, We find the lebesgue measure of the set of α whose accumulation points are all of R^d and of the complement set. In cases, where the lebesgue measure is zero, we seek finer information about Hausdorff dimensions of the corresponding sets.
Video

On the error bounds for visible points in some cut-and-project sets
Rishi Kumar, Tel Aviv University, 12 March 2026
Abstract: The density of points visible from the origin in sets such as the Ammann–Beenker point set has recently attracted attention. These sets can also be viewed as a cut-and-project set. In this talk, we will present an error estimate for the density of visible points for some class of cut-and-project sets, along with related results. Joint work with Ilya and Barak.

Absolute-winning properties of equicontinuously-twisted badly approximable points in continued fractions and beta-transformations
Jiajie Zheng, University of North Texas, 19 March 2026
Abstract: It is well know that in a $β$-transformation system for an integer \(\beta>0\), the set \(\{x: \liminf_{n\to\infty} \lvert T^nx-y_n \rvert >0\}\) has full Hausdorff dimension for all sequences \((y_n)\) in \([0,1)\) and in the Gauss map system \(\{x: \liminf_{n\to\infty} \lvert T^nx-0 \rvert >0\}\) also has full Hausdorff dimension. In this talk, I will introduce a dynamical approach to understanding these sets, and the new technique will allow us to strengthen the results so that the “targets’’ can be generalized to any equicontinuous sequence of functions, enabling the targets to vary by trajectories. In particular, notably this will imply the full dimension of non-recurrent points, bridging the problems of shrinking targets and Poincare recurrence.

Absolute-winning properties of equicontinuously-twisted badly approximable points in continued fractions and beta-transformations

Jiajie Zheng, University of North Texas, 19 March 2026

Abstract: It is well know that in a $β$-transformation system for an integer $\beta>0$, the set $\{x: \liminf_{n\to\infty} \lvert T^nx-y_n \rvert >0\}$ has full Hausdorff dimension for all sequences $(y_n)$ in $[0,1)$ and in the Gauss map system $\{x: \liminf_{n\to\infty} \lvert T^nx-0 \rvert >0\}$ also has full Hausdorff dimension. In this talk, I will introduce a dynamical approach to understanding these sets, and the new technique will allow us to strengthen the results so that the “targets’’ can be generalized to any equicontinuous sequence of functions, enabling the targets to vary by trajectories. In particular, notably this will imply the full dimension of non-recurrent points, bridging the problems of shrinking targets and Poincare recurrence.

Some bounds related to the 2-adic Littlewood conjecture vukusic
Ingrid Vukusic, University of York, 2 April 2026
Abstract: Consider alpha = (sqrt(17)-1)/8. One can check that all partial quotients in the continued fraction expansion of alpha are bounded by 3. If we multiply alpha by 2, we get a number where again all partial quotients are bounded by 3. And the same is true for 4*alpha. Might this go on forever as we keep multiplying by 2 (mod 1)? Of course, the answer is "no", as the 2-adic Littlewood conjecture is known to be true for quadratic irrationals. In this talk, we will use Hurwitz's algorithm for multiplication by 2 to approach the 2-adic Littlewood conjecture in a completely naive way, and we will (im)prove some bounds related to the 2-adic Littlewood conjecture and a variant of it. Joint work with Dinis Vitorino.In this talk, I will introduce the class of geodesically rich representations. These are representations of (real or complex) hyperbolic lattices that preserve a significant amount of the geometric structure of the associated quotient manifold. When the quotient manifold has robust geometric structure, these representations exhibit rigidity phenomena. In particular, a recent superrigidity theorem for rich representations was used to prove that finite-volume hyperbolic manifolds with infinitely many maximal totally geodesic submanifolds are arithmetic (Bader-Fisher-Miller-Stover). I will discuss a new superrigidity theorem for rich representations that efficiently recovers existing results and addresses target groups that were previously inaccessible.

Some bounds related to the 2-adic Littlewood conjecture vukusic

Ingrid Vukusic, University of York, 2 April 2026

Abstract: Consider alpha = (sqrt(17)-1)/8. One can check that all partial quotients in the continued fraction expansion of alpha are bounded by 3. If we multiply alpha by 2, we get a number where again all partial quotients are bounded by 3. And the same is true for 4*alpha. Might this go on forever as we keep multiplying by 2 (mod 1)? Of course, the answer is "no", as the 2-adic Littlewood conjecture is known to be true for quadratic irrationals. In this talk, we will use Hurwitz's algorithm for multiplication by 2 to approach the 2-adic Littlewood conjecture in a completely naive way, and we will (im)prove some bounds related to the 2-adic Littlewood conjecture and a variant of it. Joint work with Dinis Vitorino.In this talk, I will introduce the class of geodesically rich representations. These are representations of (real or complex) hyperbolic lattices that preserve a significant amount of the geometric structure of the associated quotient manifold. When the quotient manifold has robust geometric structure, these representations exhibit rigidity phenomena. In particular, a recent superrigidity theorem for rich representations was used to prove that finite-volume hyperbolic manifolds with infinitely many maximal totally geodesic submanifolds are arithmetic (Bader-Fisher-Miller-Stover). I will discuss a new superrigidity theorem for rich representations that efficiently recovers existing results and addresses target groups that were previously inaccessible.

Superrigidity of rich representations maldague
Alex Maldague, Rice University, 14 April 2026
Abstract: In this talk, I will introduce the class of geodesically rich representations. These are representations of (real or complex) hyperbolic lattices that preserve a significant amount of the geometric structure of the associated quotient manifold. When the quotient manifold has robust geometric structure, these representations exhibit rigidity phenomena. In particular, a recent superrigidity theorem for rich representations was used to prove that finite-volume hyperbolic manifolds with infinitely many maximal totally geodesic submanifolds are arithmetic (Bader-Fisher-Miller-Stover). I will discuss a new superrigidity theorem for rich representations that efficiently recovers existing results and addresses target groups that were previously inaccessible.
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Sets of exact(er) approximation order ward
Benjamin Ward, University of York, 30 April 2026
Abstract: I will present joint work with Simon Baker (Loughborough) where we introduce a quantitative notion of exactness within Diophantine approximation. Given functions Ψ : (0, ∞) → (0, ∞) and ω : (0, ∞) → (0, 1), we study the set of points that are Ψ-well approximable but not Ψ(1 − ω)-well approximable, denoted E(Ψ,ω). This generalises the set of Ψ-exact approximation order as studied by Bugeaud (Math. Ann. 2003). We prove results on the cardinality and Hausdorff dimension of E(Ψ,ω). In particular, for certain functions Ψ we find a critical threshold on ω whereby the set E(Ψ,ω) drops from positive Hausdorff dimension to empty when ω is multiplied by a constant.
Video

Fall 2025

Zero-one laws for uniform inhomogeneous Diophantine approximations
Vasiliy Neckrasov, Brandeis University, 30 September 2025
Abstract: In [Compositio Math. 155 (2019)] Kleinbock and Wadleigh proved a “zero-one law” for uniform Diophantine approximations to pairs (Θ, η) of a matrix Θ and vector η by using dynamics on the space of grids. We will show how the classical Diophantine transference principle provides an alternative approach to this problem and allows us to prove some generalizations. Namely, we will reduce the statement for pairs to the twisted (“fixed matrix”) case and show zero-one laws for twisted uniform approximations. All the proofs are made in weighted case and, more generally, in the setup of approximations with arbitrary weight functions, which will also be discussed. This talk is based on arXiv:2508.01912 and arXiv:2503.21180.
YouTube Video

Simultaneously bounded and dense orbits for commuting Cartan actions
Chengyang Wu, Peking University, 14 October 2025
Abstract: With the goal to attack Uniform Littlewood’s Conjecture proposed in [BFK25], we introduced the concept of “fiberwise nondivergence” for the action of a cone inside the full diagonal subgroup of SL₃(R). Then it is proved in our paper that there exists a dense subset of SL₃(R)/SL₃(Z) in which each point has a fiberwise non-divergent orbit under a cone inside the full diagonal subgroup and an unbounded orbit under every diagonal flow. Our proof also presented the first instance of results concerning simultaneously bounded and dense orbits for commuting actions on noncompact spaces. This is a joint work with Dmitry Kleinbock.

Simultaneously bounded and dense orbits for commuting Cartan actions

Chengyang Wu, Peking University, 14 October 2025

Abstract: With the goal to attack Uniform Littlewood’s Conjecture proposed in [BFK25], we introduced the concept of “fiberwise nondivergence” for the action of a cone inside the full diagonal subgroup of SL₃(R). Then it is proved in our paper that there exists a dense subset of SL₃(R)/SL₃(Z) in which each point has a fiberwise non-divergent orbit under a cone inside the full diagonal subgroup and an unbounded orbit under every diagonal flow. Our proof also presented the first instance of results concerning simultaneously bounded and dense orbits for commuting actions on noncompact spaces. This is a joint work with Dmitry Kleinbock.

Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications
Pratyush Sarkar, ETH, 21 October 2025
Abstract: A celebrated theorem of Eskin–Mozes–Shah gives an asymptotic counting formula for the number of integral (n x n)-matrices with a prescribed irreducible (over the integers/rationals) integral characteristic polynomial. We obtain a power saving error term for the counting problem for (3 x 3)-matrices. We do this by using the connection to homogeneous dynamics and proving effective equidistribution of translates of tori in SL₃(R)/SL₃(Z). A key tool is that the limiting Lie algebra corresponding to the translates of tori is a certain nilpotent Lie algebra. This allows us to use the recent breakthrough work of Lindenstrauss–Mohammadi–Wang–Yang on effective versions of Shah’s/Ratner’s theorems. We actually study the phenomenon more generally for any semisimple Lie group which we may discuss if time permits.

Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications

Pratyush Sarkar, ETH, 21 October 2025

Abstract: A celebrated theorem of Eskin–Mozes–Shah gives an asymptotic counting formula for the number of integral (n x n)-matrices with a prescribed irreducible (over the integers/rationals) integral characteristic polynomial. We obtain a power saving error term for the counting problem for (3 x 3)-matrices. We do this by using the connection to homogeneous dynamics and proving effective equidistribution of translates of tori in SL₃(R)/SL₃(Z). A key tool is that the limiting Lie algebra corresponding to the translates of tori is a certain nilpotent Lie algebra. This allows us to use the recent breakthrough work of Lindenstrauss–Mohammadi–Wang–Yang on effective versions of Shah’s/Ratner’s theorems. We actually study the phenomenon more generally for any semisimple Lie group which we may discuss if time permits.

The Hausdorff dimension of the intersection of ψ-well approximable numbers and self-similar sets
Suxuan Chen, Ohio State University, 28 October 2025
Abstract: Let ψ be a monotonically non-increasing function from N to R, and let ψ_v be defined by ψ_v(q)=1/q^v. Here, we consider self-similar sets whose iterated function systems satisfy the open set condition. For functions ψ that do not decrease too rapidly, we give a conjecturally sharp upper bound on the Hausdorff dimension of the intersection of ψ-well approximable numbers and such self-similar sets. When ψ=ψ_v for some v greater than 1 and sufficiently close to 1, we give a lower bound for this Hausdorff dimension, which asymptotically matches the upper bound as v approaches 1. In particular, we show that the set of very well approximable numbers has full Hausdorff dimension within self-similar sets.
Video

Ergodic theorems for dilates of submanifolds in R^d actions
Reynold Fregoli, University of Michigan, 4 November 2025
Abstract: I will discuss the validity of pointwise ergodic theorems for dilates of submanifolds in R^d-actions. In particular, I will present two recent results in this direction. The first, joint work with P. Bandi and D. Kleinbock, provides a positive result for continuous test functions in mixing R^d-actions. The second, joint work with J. Cheng and B. Guo, shows that if the regularity assumption on the test function is removed, a pointwise theorem may fail to hold.
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Pseudo-random sequences, twin primes, and twisted diophantine approximation
Manuel Hauke, TU Graz, 18 November 2025
Abstract: In this talk, I will speak about dynamics of $(a_n\alpha)_{n} \mod 1$ for integer sequences $(a_n)_n$ and fixed irrational rotations $\alpha$. The focus will be on the sequence of primes and other multiplicatively defined sequences, where gap statistics as well as twisted diophantine approximation will be considered. If time permits, I will outline the proof that includes a sieve coming from the twin prime counting problem, and establishing via random walks on Ostrowski digits an equidistribution result on diophantine Bohr sets mod d. This talk is partially based on https://arxiv.org/abs/2506.01736 and joint work with E. Kowalski https://arxiv.org/abs/2502.08335.
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Polynomial effective equidistribution for some higher dimensional unipotent subgroups
Zuo Lin, UC Berkeley, 2 December 2025
Abstract: Let G be a semisimple Lie group, Γ be a lattice in G and U be a unipotent subgroup of G. A celebrated theorem of Ratner says that for any x in G/Γ the orbit U.x is equidistributed in a periodic orbit of some subgroup U ≤ L ≤ G. Establishing a quantitative version of Ratner’s theorem has been long sought after. If U is a horospherical subgroup of G, the question is well-studied. If U is not a horospherical subgroup, this question is far less understood. Recently, Lindenstrauss, Mohammadi, Wang and Yang established a fully quantitative and effective equidistribution result for orbits of one-parameter (non-horospherical) unipotent groups in some cases. In this talk, we will discuss a recent equidistribution theorem for some unipotent subgroups in higher dimension. Our results in particular provide equidistribution theorems for orbits of the isometry group of a non-degenerate bilinear form on Rⁿ in SL_n(R)/SL_n(Z).
Video

Rectangular Shrinking Targets on Self-Similar Carpets
Demi Allen, University of Exeter, 9 December 2025
Abstract: Suppose \((X,d)\) is a metric space equipped with a Borel probability measure and suppose \((B_i)_{i \in \mathbb{N}}\) is a sequence of measurable sets in \(X\). Suppose \(T: X \to X\) is a measure preserving transformation and consider the set: \[\mathcal{B} = \{x \in X: T^n x \in B_n \text{ for infinitely many } n\in\mathbb{N}\}. \] This is a shrinking target set. The terminology of “shrinking targets” was first introduced by Hill and Velani in 1995. Since then, shrinking target problems have received a great deal of interest, especially with regards to studying the measure-theoretic and dimension-theoretic properties of shrinking target sets. In this talk, I will discuss some recent work with Thomas Jordan (Bristol, UK) and Ben Ward (York, UK) where we establish the Hausdorff dimension of a shrinking target set where our “targets” (the \(B_n\)) are rectangles and \(X\) is a self-similar carpet.

Rectangular Shrinking Targets on Self-Similar Carpets

Demi Allen, University of Exeter, 9 December 2025

Abstract: Suppose $(X,d)$ is a metric space equipped with a Borel probability measure and suppose $(B_i)_{i \in \mathbb{N}}$ is a sequence of measurable sets in $X$. Suppose $T: X \to X$ is a measure preserving transformation and consider the set: \[\mathcal{B} = \{x \in X: T^n x \in B_n \text{ for infinitely many } n\in\mathbb{N}\}. \] This is a shrinking target set. The terminology of “shrinking targets” was first introduced by Hill and Velani in 1995. Since then, shrinking target problems have received a great deal of interest, especially with regards to studying the measure-theoretic and dimension-theoretic properties of shrinking target sets. In this talk, I will discuss some recent work with Thomas Jordan (Bristol, UK) and Ben Ward (York, UK) where we establish the Hausdorff dimension of a shrinking target set where our “targets” (the $B_n$) are rectangles and $X$ is a self-similar carpet.

Spring 2025

Khintchine’s theorem on self-similar measures on the real line
Han Zhang, 11 February 2025
Abstract: In 1984, Mahler proposed the following question on Diophantine approximation : How close can irrational numbers in the middle-thirds Cantor set be approximated by rational numbers? One way to reformulate Mahler’s question is to ask if Khintchine’s theorem extends to the middle-thirds Cantor set. In a joint work with Timothée Bénard and Weikun He, we prove that Khintchine’s theorem holds for any self-similar measures on the real line. In particular this applies to the Hausdorff measure on the middle-thirds Cantor set. Our result generalizes the recent breakthrough work of Khalil-Luethi in dimension one. Our proof is inspired by the work of Bénard-He regarding the semisimple random walks on homogeneous spaces.

Khintchine’s theorem on self-similar measures on the real line

Han Zhang, 11 February 2025

Abstract: In 1984, Mahler proposed the following question on Diophantine approximation : How close can irrational numbers in the middle-thirds Cantor set be approximated by rational numbers? One way to reformulate Mahler’s question is to ask if Khintchine’s theorem extends to the middle-thirds Cantor set. In a joint work with Timothée Bénard and Weikun He, we prove that Khintchine’s theorem holds for any self-similar measures on the real line. In particular this applies to the Hausdorff measure on the middle-thirds Cantor set. Our result generalizes the recent breakthrough work of Khalil-Luethi in dimension one. Our proof is inspired by the work of Bénard-He regarding the semisimple random walks on homogeneous spaces.

Some geometric and dynamical properties of hyperbolic magnetic flows
JinCheng Wang, 4 March 2025
Abstract: As a generalization of geodesic flows, magnetic flows trace unit-speed curves with constant geodesic curvature. We consider the magnetic flows of surfaces with negative Gaussian curvature that are nonuniformly hyperbolic. By studying its geometry on the universal covering of the surface, we show the uniqueness of the measure of maximal entropy via the Bowen-Climenhaga-Thompson machinery. This is a joint work with Boris Hasselblatt.

Some geometric and dynamical properties of hyperbolic magnetic flows

JinCheng Wang, 4 March 2025

Abstract: As a generalization of geodesic flows, magnetic flows trace unit-speed curves with constant geodesic curvature. We consider the magnetic flows of surfaces with negative Gaussian curvature that are nonuniformly hyperbolic. By studying its geometry on the universal covering of the surface, we show the uniqueness of the measure of maximal entropy via the Bowen-Climenhaga-Thompson machinery. This is a joint work with Boris Hasselblatt.

Equidistribution of polynomially bounded o-minimal curves in homogenous spaces
Michael Bersudsky, 11 March 2025
Abstract: I will present my recent joint work with Nimish Shah and Hao Xing. We study a basic question about the limiting distribution of averages along unbounded curves in homogeneous spaces. We consider the class of curves which are definable in o-minimal structures, for example, matrix curves in SL(n,R) who’s entries are rational functions. Our results extend Ratner’s equidistribution theorem for one-parameter unipotent flows, generalize Shah’s results for polynomial curves and generalize some of the recent results Peterzil and Starchenko. A key component in our work is the Kleinbock-Margulis $(C,α)$-good property for families of functions definable in polynomially bounded o-minimal structures. The property for such functions was unknown before, and its proof relies on the tools of o-minimal structures theory.
Video

The Dirichlet spectrum
Alon Agin, 18 March 2025
Abstract: Akhunzhanov and Shatskov defined the Dirichlet spectrum, corresponding to mxn matrices and to norms on R^m and Rⁿ. In case (m,n) = (2,1) and using the Euclidean norm on R², they showed that the spectrum is an interval. We generalize this result to arbitrary (m,n) with max(m,n)>1 and arbitrary norms, improving previous works from recent years. We also define some related spectra and show that they too are intervals. We also prove the existence of matrices exhibiting special properties with respect to their uniform exponent. Our argument is a modification of an argument of Khintchine from 1926.
Video

Weighted Inhomogeneous Bad is Winning and Null
Liyang Shao, 22 April 2025
Abstract: We will introduce the notion of inhomogeneous weighted badly approximable vectors. We discuss that this set can be very large (winning) in a sense and in some other sense it is very small (measure wise). In particular, we talk about such largeness and smallness via studying weighted inhomogeneous bad intersected with manifolds and support of certain measures. This is a joint work with Shreyasi Datta.

Weighted Inhomogeneous Bad is Winning and Null

Liyang Shao, 22 April 2025

Abstract: We will introduce the notion of inhomogeneous weighted badly approximable vectors. We discuss that this set can be very large (winning) in a sense and in some other sense it is very small (measure wise). In particular, we talk about such largeness and smallness via studying weighted inhomogeneous bad intersected with manifolds and support of certain measures. This is a joint work with Shreyasi Datta.

Random Walks on SL₂(F_p) x SL₂(F_p)
Srivatsa Srinivas, 29 April 2025
Abstract: We will give a taste of the flavors of math that constitute the study of random walks on compact groups, followed by which we will describe the author’s work with Prof. Golsefidy in solving a question of Lindenstrauss and Varju. Namely, can the spectral gap of a random walk on a product of groups be related to those of the projections onto its factors
Video

Higher rank Furstenberg slicing
Emilio Corso, 6 May 2025
Abstract: In addition to the notoriously challenging measure rigidity conjecture for non-lacunary semigroups of toral endomorphisms, Furstenberg formulated, in the late sixties, a series of geometric rigidity conjectures aimed, as their dynamical counterparts, at capturing the heuristic principle that expansions of real numbers in multiplicatively independent integer bases are uncorrelated. Among these features the intersection conjecture, according to which the intersection of closed sets invariant under multiplicatively independent toral endomorphisms is, in dimensional terms, as small as it can be given the constraints of the system. The conjecture was settled by Shmerkin half a century after its formulation, with subsequent alternative arguments given by Meng Wu and Tim Austin. In joint work with Shmerkin, we provide a higher rank extension of the result, which handles intersections of any finite collection of invariant sets, and more generally presents uniform dimensional estimates for slices of products of such sets with arbitrary affine subspaces. As in the rank-one case, the argument hinges upon a precise understanding of Frostman exponents, via L^q spectra, for convolutions of self-similar measures.
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Fall 2024

Smooth discrepancy and Littlewood’s conjecture
Sam Chow, 24 September 2024
Abstract: Given \boldsymbol α ∈ [0,1]^d, we estimate the smooth discrepancy of the Kronecker sequence (n \boldsymbol α \: \mathrm{mod} \: 1)_n=1^∞. We find that it can be smaller than the classical discrepancy of any sequence when d ≤ 2, and can even be bounded in the case d=1. To achieve this, we establish a novel deterministic analogue of Beck’s local-to-global principle (Annals 1994), which relates the discrepancy of a Kronecker sequence to multiplicative diophantine approximation. This opens up a new avenue of attack for Littlewood’s conjecture.
Video

Metric Diophantine approximation: Moving targets and inhomogeneous variants
Manuel Hauke, 8 October 2024
Abstract: Khintchine’s Theorem and its inhomogeneous and multidimensional variants provide a satisfying answer about the quality of approximations for almost every number. In this talk, I will discuss the (still open) question of allowing a moving target (that is, the inhomogeneous parameter changes for each denominator) in Khintchine’s Theorem. Furthermore, I will describe Duffin–Schaeffer-type results and conjectures in these setups, both in dimension 1, but also in higher dimensions. This is partially joint work with Victor Beresnevich and Sanju Velani, respectively with Felipe Ramírez.
Video

Fourier Asymptotics and Effective Equidistribution
Shreyasi Datta, 5 November 2024
Abstract: We talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics. This is a joint work with Subhajit Jana.
Video

Spectral independence of compact groups
Keivan Mallahi-Karai, 19 November 2024
Abstract: Let $G_1$ and $G_2$ be compact simple (real or $p$-adic) Lie groups, and let $\mu_1$ and $\mu_2$ be symmetric probability measures on $G_1$ and $G_2$. Under mild conditions on $\mu_1$ and $\mu_2$, the distribution of $\mu_i$ random walks on $G_i$ converges to the uniform measure, and the speed of convergence is governed by the spectral gap. A coupling of $\mu_1$ and $\mu_2$ is any probability measure $\mu$ on $G_1 \times G_2$ whose marginal distributions are $\mu_1$ and $\mu_2$, respectively . A natural question is under what conditions a spectral gap for all couplings depending on spectral gaps of $\mu_1$ and $\mu_2$ can be established. In this talk, I will present results in this direction which are based on joint work with Alireza S. Golsefidy and Amir Mohammadi.
Video

Singular matrices on fractals
Gaurav Aggarwal, 3 December 2024
Abstract:* Singular vectors are those for which Dirichlet’s Theorem can be improved by arbitrarily small multiplicative constants. Recently, Kleinbock and Weiss showed that the set of singular vectors has measure zero with respect to any friendly measure. However, determining their Hausdorff dimension remains a subtle and challenging problem. Khalil addressed this by proving that the Hausdorff dimension of the set of singular vectors intersecting a self-similar fractal is strictly smaller than the fractal’s dimension. In this talk, I will extend Khalil’s result in four key directions. First, we generalize the study from vectors to matrices. Second, we analyze intersections with products of fractals, such as the Cartesian product of the middle-third and middle-fifth Cantor sets. Third, we establish upper bounds for singular vectors in a generalized weighted setting. Finally, we derive an upper bound on the Hausdorff dimension of $ω$-very singular matrices in these broader settings, extending earlier work of Das, Fishman, Simmons, and Urbanski, who studied the real, unweighted case. Our approach is dynamical in nature, relying on the construction of a height function inspired by the work of Kadyrov, Kleinbock, Lindenstrauss, and Margulis. This is a joint work with Anish Ghosh.
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Escape of Mass of Sequences
Noy Soffer Aranov, 10 December 2024
Abstract: One way to study the distribution of nested quadratic number fields satisfying fixed arithmetic relationships is through the evolution of continued fraction expansions. In the function field setting, it was shown by de Mathan and Teullie that given a quadratic irrational $\Theta$, the degrees of the periodic part of the continued fraction of $t^n\Theta$ are unbounded. Paulin and Shapira improved this by proving that quadratic irrationals exhibit partial escape of mass. Moreover, they conjectured that they must exhibit full escape of mass. We construct counterexamples to their conjecture in every characteristic. In this talk we shall discuss the technique of proof as well as the connection between escape of mass in continued fractions, Hecke trees, and number walls. This is part of ongoing works with Erez Nesharim and with Steven Robertson.
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Spring 2024

Incidence geometry and effective equidistribution in homogeneous dynamics
Lei Yang, 29 February 2024
Abstract: I will explain my proof of an effective version of Ratner’s equidistribution theorem for unipotent orbits in SL(3,R)/SL(3,Z). The proof combines new ideas from harmonic analysis and incidence geometry. In particular, the proof is based on a bootstrapping argument improving the local dimension of measures generated by unipotent orbits. The key is to relate the behavior of the unipotent orbits to a Kakeya model.
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Joint Equidistribution of Approximates
Gaurav Aggarwal, 7 March 2024
Abstract: The distribution of integer points on varieties has occupied mathematicians for centuries. In the 1950’s Linnik used an “ergodic method” to prove the equidistribution of integer points on large spheres under a congruence condition. As shown by Maaß, this problem is closely related to modular forms. Subsequently, there were spectacular developments both from the analytic as well as ergodic side. I will discuss a more refined problem, namely the joint distribution of lattice points in conjunction with other arithmetic data. An example of such data is the “shape” of an associated lattice, or in number theoretic language, a Heegner point. In a completely different direction, a “Poincaré section” is a classical and useful tool in ergodic theory and dynamical systems. Recently, Shapira and Weiss, constructed a Poincaré section for the geodesic flow on the moduli space of lattices to study joint equidistribution properties. Their work in fact is very general but crucially uses the fact that the acting group has rank one. In joint work with Anish Ghosh, we develop a new method to deal with actions of higher rank groups. I will explain this and, if time permits, some corollaries in Diophantine analysis.
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Bad is null via constant invariance
Shreyasi Datta, 14 March 2024
Abstract: The set of badly approximable vectors in Diophantine approximation plays a significant role. In a recent work with Victor Beresnevich, Anish Ghosh, and Ben Ward, we developed a general framework to show a `constant invariance’ property for a large class of limsup sets of neighbourhoods of subsets of a metric measure space. As a consequence, we get that the set of badly approximable points has measure zero in a metric space equipped with certain natural measures. In particular, given any C² manifold, we show almost every point is not badly approximable.
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Quantum ergodicity on the Bruhat-Tits building for PGL(3) in the Benjamini-Schramm limit
Carsten Peterson, 11 April 2024
Abstract: Originally, quantum ergodicity concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic. More recently, several authors have investigated quantum ergodicity for sequences of spaces which “converge” to their common universal cover and when one restricts to eigenfunctions with eigenvalues in a fixed range. Previous authors have considered this type of quantum ergodicity in the settings of regular graphs, rank one symmetric spaces, and some higher rank symmetric spaces. We prove analogous results in the case when the underlying common universal cover is the Bruhat-Tits building associated to PGL(3, F) where F is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting.
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Bounded ratios and badly approximability
Nikolay Moshchevitin, 18 April 2024
Abstract: We will discuss relatively new criteria of badly approximability in terms of ratios of best approximations. Let \(q_\nu\) be convergents of continued fractions to real irrational \(\alpha\). It is well known that \[\alpha\textrm{ is badly approximable }\iff\sup_\nu \frac{q_{\nu+1}}{q_\nu}\textrm{ is finite }\iff\inf_\nu\frac{\lVert q_{\nu+1}\alpha \rVert}{\lVert q_{\nu}\alpha \rVert} >0.\] We will discuss how this property may be generalised to Diophantine Approximation in higher dimensions. The answer seems to be rather non-trivial. Some of the related properties may be expressed in terms of Parametric Geometry of Numbers recently developed by Schmidt, Summerer, Roy and the others. Also we discuss some properties of ratios under the consideration in accordance with the study of multidimensional Dirichlet spectra.

Bounded ratios and badly approximability

Nikolay Moshchevitin, 18 April 2024

Abstract: We will discuss relatively new criteria of badly approximability in terms of ratios of best approximations. Let $q_\nu$ be convergents of continued fractions to real irrational $\alpha$. It is well known that \[\alpha\textrm{ is badly approximable }\iff\sup_\nu \frac{q_{\nu+1}}{q_\nu}\textrm{ is finite }\iff\inf_\nu\frac{\lVert q_{\nu+1}\alpha \rVert}{\lVert q_{\nu}\alpha \rVert} >0.\] We will discuss how this property may be generalised to Diophantine Approximation in higher dimensions. The answer seems to be rather non-trivial. Some of the related properties may be expressed in terms of Parametric Geometry of Numbers recently developed by Schmidt, Summerer, Roy and the others. Also we discuss some properties of ratios under the consideration in accordance with the study of multidimensional Dirichlet spectra.

Flexibility and rigidity for Cantor repellers
Alena Erchenko, 2 May 2024
Abstract: We will consider dynamical systems that we call Cantor repellers which are expanding maps on invariant Cantor sets coming from iterated function systems. Cantor repellers have two natural invariant measures: the measure of full dimension and the measure of maximal entropy. We show that dimensions and Lyapunov exponents of those measures are flexible up to well understood restrictions. We will also discuss the boundary case for the range of values of the considered dynamical data. This is joint work with Jacob Mazor.
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Fall 2023

Birkhoff generic points on curves
Omri Solan, 21 September 2023
Abstract: Let a_t be a diagonal flow on the space X of unimodular lattices in Rⁿ. A point x in X is called Birkhoff generic if a_t.x equidistributes in X as t→ ∞. By Birkhoff ergodic theorem, almost every point x in X is Birkhoff generic. One may ask whether the same is true when the point x is sampled according to a measure singular to Lebesgue. In a joint work with Andreas Wieser, we discuss the case of a generic point x in an analytic curve in X, and show that under certain conditions, it must be Birkhoff generic. This Birkhoff genericity result has various applications in Diophantine approximation. In this talk we will relate Birkhoff genericity to approximations of real numbers by algebraic numbers of degree at most n.
Video

Stochastic Calculus for the Theta Process
Zach Selk, 28 September 2023
Abstract: The Theta process, $X(t)$, is a complex valued stochastic process of number theoretical origin arising as a scaling limit of quadratic Weyl sums \[\sum_{n=1}^N e^{2\pi i \left(\frac{1}{2}(n^2+\beta)x+\alpha n\right)},\] where $(\alpha,\beta)\in \mathbb R^2 \setminus \mathbb Q^2$ and $x\in \mathbb R$ is chosen at random according to any law absolutely continuous with respect to Lebesgue measure. The Theta process can be explicitly represented as $X(t)=\sqrt{t} \Theta(\Gamma g \Phi^{2 \log t})$ where $\Theta$ is an automorphic function defined on Lie group $G$, invariant under left multiplication under lattice $\Gamma$. Additionally, $g\in \Gamma \setminus G$ is chosen Haar uniformly at random and $\Phi$ is the geodesic flow on $\Gamma \setminus G$. The Theta process shares several similar properties with the Brownian motion. In particular, both lack differentiability and have the same $p$ variation and H\”older properties. Similarly to Brownian motion, standard calculus and even Young/Riemann-Stieltjes calculus techniques do not work. However, Brownian motion is what is known as a martingale allowing for a classical theory of It\^o calculus which makes use of cancellations “on average”. The It\^o calculus can be used to prove several properties of Brownian motion such as its conformal invariance, bounds on its running maximum in terms of its quadratic variation, absolutely continuous changes in measure and much more. Unfortunately, we show that the Theta process $X$ is not a (semi)martingale, therefore It\^o techniques don’t work. However, a new theory introduced in 1998 by Terry Lyons called rough paths theory handles processes with the same analytic regularity as $X$. The key idea in rough paths theory is that constructing stochastic calculus for a signal can be reduced to constructing the “iterated integrals” of the signal. In this talk, we will show the construction of the iterated integrals – the “rough path” – above the process $X$. Joint with Francesco Cellarosi.
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A variation on the p-adic Littlewood Conjecture
Yuval Yifrach, 5 October 2023
Abstract: We consider a variation on the p-adic Littlewood Conjecture where instead of using powers of one prime, we use arbitrarily large primes. We examine this conjecture from two view points: the Diophantine-approximation viewpoint and the dynamical viewpoint. Using the dynamical view point, we rephrase the conjecture in terms of Hecke neighbors and prove partial results towards the conjecture. Namely, we prove that the Hausdorff dimension of certain exception sets is strictly smaller than 1. In addition, we show that the conjecture holds in the quadratic irrational case, and that it holds for a generic numbers according to a family of measures supported on a Lebesgue null sets. Our tools for the proof are mainly the effective equidistribution of Hecke neighbors due to Oh Et. Al. and to expander properties of SL₂(Z/pZ) due to Bourgain-Gamburd. This talk is based on a joint work with Erez Nesharim.

A variation on the p-adic Littlewood Conjecture

Yuval Yifrach, 5 October 2023

Abstract: We consider a variation on the p-adic Littlewood Conjecture where instead of using powers of one prime, we use arbitrarily large primes. We examine this conjecture from two view points: the Diophantine-approximation viewpoint and the dynamical viewpoint. Using the dynamical view point, we rephrase the conjecture in terms of Hecke neighbors and prove partial results towards the conjecture. Namely, we prove that the Hausdorff dimension of certain exception sets is strictly smaller than 1. In addition, we show that the conjecture holds in the quadratic irrational case, and that it holds for a generic numbers according to a family of measures supported on a Lebesgue null sets. Our tools for the proof are mainly the effective equidistribution of Hecke neighbors due to Oh Et. Al. and to expander properties of SL₂(Z/pZ) due to Bourgain-Gamburd. This talk is based on a joint work with Erez Nesharim.

Equidistribution problem in the space of Euclidean sublattices
Hao Xing, 19 October 2023
Abstract: Consider the space of covolume-one sublattices of a fixed rank m in the Euclidean space ℝd. How do the orbits behave under the action of the lattice subgroups of SL(d,ℝ) (e.g. SL(d,ℤ))? In a recent joint work with Michael Bersudsky, we established an equidistribution phenomenon of such orbits when d=m+1. However, there are many more unsolved problems along this direction which might be of interest not only to homogeneous dynamicists, but also to number theorists and analysts as well. In this talk, I will explain the problem, our result, an overview of methods and further directions of research in a user-friendly way.

Equidistribution problem in the space of Euclidean sublattices

Hao Xing, 19 October 2023

Abstract: Consider the space of covolume-one sublattices of a fixed rank m in the Euclidean space ℝd. How do the orbits behave under the action of the lattice subgroups of SL(d,ℝ) (e.g. SL(d,ℤ))? In a recent joint work with Michael Bersudsky, we established an equidistribution phenomenon of such orbits when d=m+1. However, there are many more unsolved problems along this direction which might be of interest not only to homogeneous dynamicists, but also to number theorists and analysts as well. In this talk, I will explain the problem, our result, an overview of methods and further directions of research in a user-friendly way.

Covering Radii in Positive Characteristic
Noy Soffer Aranov, 26 October 2023
Abstract: A fascinating question in geometry of number pertains to the covering radius of lattice with respect to an interesting function. For example, given a convex body C and a lattice L in R^d, it is interesting to ask what is the infimal r ≥ 0 such that L + rC = R^d. Another interesting covering radius is the multiplicative covering radius, which connects to dynamics due to its invariance under the diagonal group. It was conjectured by Minkowski that the multiplicative covering radius is bounded above by 2^-d and that this upper bound is obtained only on AZ^d. In this talk I will discuss surprising results pertaining to covering radii in the positive characteristic setting and discover several surprising results. Some of my results include explicitly connecting between the covering radii with respect to convex bodies and successive minima and proving a positive characteristic analogue of Minkowski’s function.
Video

Constructing best approximation vectors
Alon Agin, 2 November 2023
Abstract: For v in R^d and arbitrary norm, we define the best approximation sequence of v and the displacement vectors sequence of v. We will discuss classical and recent works in Diophantine approximations in the language of these objects – focusing on their length, direction and congruence class.
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Dispersion and Littlewood’s conjecture
Sam Chow, 9 November 2023
Abstract: I’ll discuss some problems related to Littlewood’s conjecture in diophantine approximation, and the role hitherto played by discrepancy theory. I’ll explain why our new dispersion-theoretic approach should, and does, deliver stronger results. Our dispersion estimate is proved using Poisson summation and diophantine inequalities. Joint with Niclas Technau.
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Jordan and Cartan spectra in higher rank with applications to correlations
Mikey Chow, 16 November 2023
Abstract: The celebrated prime geodesic theorem for a closed hyperbolic surface says that the number of closed geodesics of length at most t is asymptotically e^t/t. For a closed surface equipped with two different hyperbolic structures, Schwartz and Sharp (’93) showed that the number of free homotopy classes of length about t in both hyperbolic structures is asymptotically a constant multiple of e^ct /t^3/2 for some 0<c<1. We will discuss the asymptotic correlations of the length spectra of convex cocompact manifolds, generalizing Schwartz-Sharp’s results. Surprisingly, it is helpful for us to relate this problem with understanding the Jordan spectrum of a discrete subgroup in higher rank. In particular, we will explain the source of the exponential and polynomial factors in Schwartz-Sharp’s asymptotics from a higher rank viewpoint. We will also discuss the asymptotic correlations of the displacement spectra and the ratio law between the asymptotic correlations of the length and displacement spectra. This is joint work with Hee Oh.

Jordan and Cartan spectra in higher rank with applications to correlations

Mikey Chow, 16 November 2023

Abstract: The celebrated prime geodesic theorem for a closed hyperbolic surface says that the number of closed geodesics of length at most t is asymptotically e^t/t. For a closed surface equipped with two different hyperbolic structures, Schwartz and Sharp (’93) showed that the number of free homotopy classes of length about t in both hyperbolic structures is asymptotically a constant multiple of e^ct /t^3/2 for some 0<c<1. We will discuss the asymptotic correlations of the length spectra of convex cocompact manifolds, generalizing Schwartz-Sharp’s results. Surprisingly, it is helpful for us to relate this problem with understanding the Jordan spectrum of a discrete subgroup in higher rank. In particular, we will explain the source of the exponential and polynomial factors in Schwartz-Sharp’s asymptotics from a higher rank viewpoint. We will also discuss the asymptotic correlations of the displacement spectra and the ratio law between the asymptotic correlations of the length and displacement spectra. This is joint work with Hee Oh.

Primitive rational points on expanding horospheres: effective joint equidistribution
Daniel El-Baz, 30 November 2023
Abstract: I will present joint work with Min Lee and Andreas Strömbergsson. Using techniques from analytic number theory, spectral theory, geometry of numbers as well as a healthy dose of linear algebra and building on a previous work by Bingrong Huang, Min Lee and myself, we furnish a new proof of a 2016 theorem by Einsiedler, Mozes, Shah and Shapira. That theorem concerns the equidistribution of primitive rational points on certain homogeneous spaces and our proof has the added benefit of yielding a rate of convergence. It turns out to have several (perhaps surprising) applications to number theory and combinatorics, which I shall also discuss.

Primitive rational points on expanding horospheres: effective joint equidistribution

Daniel El-Baz, 30 November 2023

Abstract: I will present joint work with Min Lee and Andreas Strömbergsson. Using techniques from analytic number theory, spectral theory, geometry of numbers as well as a healthy dose of linear algebra and building on a previous work by Bingrong Huang, Min Lee and myself, we furnish a new proof of a 2016 theorem by Einsiedler, Mozes, Shah and Shapira. That theorem concerns the equidistribution of primitive rational points on certain homogeneous spaces and our proof has the added benefit of yielding a rate of convergence. It turns out to have several (perhaps surprising) applications to number theory and combinatorics, which I shall also discuss.

Finding Infinite arithmetic structures in sets of positive density
Florian Richter, 7 December 2023
Abstract: In the 1970’s Erdos asked several questions about what kind of infinite arithmetic structures can be found in every set of natural numbers with positive density. In recent joint work with Bryna Kra, Joel Moreira, and Donald Robertson we use ergodic methods to resolve some of these long-standing conjectures. This talk will provide a gentle introduction into this topic, with an overview of our results and the dynamical structures that are used to prove them.

Finding Infinite arithmetic structures in sets of positive density

Florian Richter, 7 December 2023

Abstract: In the 1970’s Erdos asked several questions about what kind of infinite arithmetic structures can be found in every set of natural numbers with positive density. In recent joint work with Bryna Kra, Joel Moreira, and Donald Robertson we use ergodic methods to resolve some of these long-standing conjectures. This talk will provide a gentle introduction into this topic, with an overview of our results and the dynamical structures that are used to prove them.

Partition regularity of Pythagorean pairs
Oleksiy Klurman, 14 December 2023
Abstract: Is there a partition of the natural numbers into finitely many pieces, none of which contains a Pythagorean triple (i.e. a solution to the equation x2+y2=z2)? This is one of the simplest (to state!) questions in arithmetic Ramsey theory which is still widely open. I will talk about a recent partial result, showing that “Pythagorean pairs” are partition regular, that is in any finite partition of the natural numbers there are two numbers x,y in the same cell of the partition, such that x2+y2=z2 for some integer z (which may be coloured differently). The proof is a blend of ideas from ergodic theory and multiplicative number theory. Based on a joint work with N. Frantzikinakis and J. Moreira.

Partition regularity of Pythagorean pairs

Oleksiy Klurman, 14 December 2023

Abstract: Is there a partition of the natural numbers into finitely many pieces, none of which contains a Pythagorean triple (i.e. a solution to the equation x2+y2=z2)? This is one of the simplest (to state!) questions in arithmetic Ramsey theory which is still widely open. I will talk about a recent partial result, showing that “Pythagorean pairs” are partition regular, that is in any finite partition of the natural numbers there are two numbers x,y in the same cell of the partition, such that x2+y2=z2 for some integer z (which may be coloured differently). The proof is a blend of ideas from ergodic theory and multiplicative number theory. Based on a joint work with N. Frantzikinakis and J. Moreira.

Fall 2022

Overlapping iterated function systems from the perspective of Metric Number Theory
Simon Baker, 22 September 2022
Abstract: Khintchine’s theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the divergence of naturally occurring volume sums. Importantly this result provides a quantitative description of how the rationals are distributed within the reals. In this talk I will discuss some recent work where I prove that a similar Khintchine like phenomenon occurs typically within many families of overlapping iterated function systems. Families of iterated function systems these results apply to include those arising from Bernoulli convolutions, the 0,1,3 problem, and affine contractions with varying translation parameters. Time permitting I also will discuss a particular family of iterated function systems for which we can be more precise. Our analysis of this family shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures.
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An avoidance principle and Margulis functions for expanding translates of unipotent orbits
Juno Seong, 6 October 2022
Abstract: Avoidance principles — quantifying how much time trajectories avoid certain subsets of the ambient space — have been fruitful in the study of dynamical systems. We prove an avoidance principle for expanding translates of unipotent orbits for some semisimple homogeneous spaces. In addition, we prove a quantitative isolation result of closed orbits and give an upper bound on the number of closed orbits of bounded volume. The proof of our results relies on the construction of a Margulis function and the theory of finite dimensional representations of semisimple Lie groups. This is joint work with Anthony Sanchez.
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A Spectral Approach to Counting and Equidistribution
Christopher Lutsko, 13 October 2022
Abstract: Since the early 20th century, spectral methods have been used to obtain effective counting theorems for various objects of interest in number theory, geometry and group theory. In this talk I’ll start by introducing two classical problems: the Gauss circle problem, and the Apollonian counting problem. By surveying results on these problems (and some generalizations), I’ll demonstrate how to use spectral methods to obtain effective asymptotics for some very classical problems. Then I will try and explain how to generalize this method to apply to certain horospherical equidistribution theorems.
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Short closed geodesics in higher rank arithmetic locally symmetric spaces
Lam Pham, 20 October 2022
Abstract: A well-known conjecture of Margulis predicts that there is a uniform lower bound on the systole of any irreducible arithmetic locally symmetric space. Recently, in joint work with Mikolaj Fraczyk, we show that for simple Lie groups of higher rank, this conjecture is equivalent to a well-known conjecture in number theory: that Salem numbers are uniformly bounded away from 1. I will discuss our proof and some tools used, and some additional results which hold unconditionally and highlight the structure of the bottom of the length spectrum.
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p-Adic Diophantine approximation with respect to fractal measures
Shreyasi Datta, 3 November 2022
Abstract: In a recent work with Anish Ghosh and Victor Beresnevich we solved a conjecture of Kleinbock and Tomanov, which shows pushforward of a p-adic fractal measure by ‘nice’ functions exhibits ‘nice’ Diophantine properties. In particular, we prove p-adic analogue of a result by Kleinbock, Lindenstrauss and Weiss on friendly measures. I will talk about how lack of the mean value theorem makes life difficult in the p-adic fields, and how we can sometimes overcome this problem.
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On inhomogeneous Diophantine approximation
Nikolay Moshchevitin, 10 November 2022
Abstract: We will discuss some classical and modern results related to systems of inhomogeneous linear forms. We will begin with Kronecker approximation theorem and famous results by Khintchine and continue with rather modern problems, in particular related to weighted setting and coprime approximation.
Video

Rational approximations to linear subspaces
Nicolas de Saxce, 17 November 2022
Abstract: Using diagonal orbits on the space of lattices, we revisit some old questions of Schmidt concerning diophantine approximation on Grassmann varieties, and in particular, we prove a version of Dirichlet’s principle in that setting.

Tail Asymptotics for Generalised Theta Sums with Rational Parameters
Tariq Osman, 1 December 2022
Abstract: We define generalised theta sums as exponential sums of the form S^f_N(x; α, β) := ∑_{n ∈ \mathbb Z} f(n/N) e((1/2 n² + β n)x + α n), where e(z) = e^{2 π i z}. If α and β are fixed real numbers, and x is chosen randomly from the unit interval, we may use homogeneous dynamics to show that N^-1/2 S^f_N$ possesses a limiting distribution as N goes to infinity, provided f is sufficiently regular. In joint work with F. Cellarosi, we prove that for specific rational pairs (α, β) this limiting distribution is compactly supported and that all other rational pairs lead to a limiting distribution with heavy tails. This complements the existing work of F. Cellarosi and J. Marklof where at least one of α or β is irrational.
Video

Counting rationals and diophantine approximation on fractals
Sam Chow, 8 December 2022
Abstract: We count rationals in missing-digit sets, with applications to diophantine approximation. In the process, we develop the theory of Fourier ℓ¹ dimension, including the computational aspect.

Spring 2022

An adelic version of the three gap theorem
Akshat Das, 3 February 2022
Abstract: In order to understand problems in dynamics which are sensitive to arithmetic properties of return times to regions, it is desirable to generalize classical results about rotations on the circle to the setting of rotations on adelic tori. One such result is the classical three gap theorem, which is also referred to as the three distance theorem and as the Steinhaus problem. It states that, for any real number, a, and positive integer, N, the collection of points na mod 1, where n runs from 1 to N, partitions the circle into component arcs having one of at most three distinct lengths. Since the 1950s, when this theorem was first proved independently by multiple authors, it has been reproved numerous times and generalized in many ways. One of the more recent proofs has been given by Marklof and Strömbergsson using a lattice based approach to gaps problems in Diophantine approximation. In this talk, we use an adaptation of this approach to the adeles to prove a natural generalization of the classical three gap theorem for rotations on adelic tori. This is joint work with Alan Haynes.
Video

The asymptotic distribution of the joint values of the integral lattice points for a system of a quadratic form and a linear form

Jiyoung Han, 10 February 2022

Abstract: Let Q be a quadratic form and let L be a linear form on the n-dimensional real vector space. We are interested in the distribution of the image of the integral lattice under the map (Q, L). Developing the celebrated work of Eskin, Margulis, and Mozes in 1998, we provide the conditions of systems of forms which satisfy that the number of integral vectors in the ball of radius T whose joint values are contained in a given bounded set converges asymptotically to the volume of the region given by the level sets of the quadratic form and the linear form, intersecting with the ball of radius T, as T goes to infinity. This condition is introduced by Gorodnik in 2004. For this, we need to classify all intermediate subgroups between the special orthogonal group preserving Q and L and the special linear group. Among them, only two closed subgroups are of our concern. We will introduce Siegel integral formulas and equidistribution theorems for each subgroup, and show how to reach our main theorem. This is joint work with Seonhee Lim and Keivan Mallahi-Karai.

Limit laws in the lattice counting problem. The case of ellipses

Julien Trevisan, 17 February 2022

Abstract: Let E be an ellipse centered around 0. We are interested in the asymptotic distribution of the error of the number of unimodular lattice points that fall into tE when the lattice is random and when t goes to infinity. Building on previous works by Bleher and by Fayad and Dolgopyat, we show that the error term, when normalized by the square root of t, converges in distribution towards an explicit distribution. For this, we first use harmonic analysis to reduce the study of the normalized error to the study of a Siegel transform that depends on t. Then, and this is the key part of our proof, we show that, when t goes to infinity, this last Siegel transform behaves in distribution as, what we call, a modified Siegel transform with random weights. Such objects often appear in average counting problems. Finally, we show that this last quantity converges almost surely, and we study the existence of the moments of its law. This work was supervised by Bassam Fayad.

S-adic quadratic forms and homogeneous dynamics
Irving Calderón, 3 March 2022
Abstract: We present two new quantitative results about quadratic forms. Let S = {∞} ∪ Sf be a finite set of places of Q. Consider the ring ZS of S-integers, and QS = ∏{p ∈ S} Qp. The first is a solution to the problem of deciding if any given integral quadratic forms Q1 and Q2 are ZS-equivalent. The proof is based on a reformulation of the problem in terms of the action of O(Q1, QS) on the space X{d,S} of lattices of QSd. A key tool are explicit mixing rates for the action of O(Q1, QS) on closed orbits in X{d,S}. As an application we obtain, for any S-integral orthogonal group, polynomial bounds on the S-norms of the elements of a finite generating set. These two results and the methods of proof are based on the work of H. Li and G. Margulis for S = { ∞ }.
Video

Classification of divergence of trajectories
Nattalie Tamam, 10 March 2022
Abstract: As shown by Dani, diophantine approximations are in direct correspondence to the behavior of orbits in certain homogeneous spaces. We will discuss the interpretation of the divergent trajectories and the obvious ones, the ones diverging due to a purely algebraic reason. As conjectured by Barak Weiss, there is a complete classification of divergent trajectories when considering the action of subgroups of the diagonal group. We will discuss the last part of this conjecture, showing that for a ‘large enough’ such subgroup, every divergent trajectory diverges obviously. This is a joint work with Omri Solan.
Video

Effective Counting and Spiralling of Lattice Approximates

Nate Hughes, 17 March 2022

Abstract: We will prove an effective version of Dirichlet’s approximation theorem, giving the error between the number of rational approximations to a real vector with denominator less than some real number T and the asymptotic growth of this count. Additional results for linear forms can be obtained, as well as results measuring the direction of these approximates, known as ‘spiralling of lattice approximates’. These results are obtained by reformulating the number-theoretic problem to the context of homogeneous spaces of unimodular lattices. The advantage of this reformulation is that we have more tools to deal with the problem, such as Siegel’s mean value theorem and Rogers’ higher moment formula. The proof involves using the ergodic properties of diagonal flows on this homogeneous space to calculate the number of lattice approximates, bounding the second moment of the count, then applying an effective ergodic theorem due to Gaposhkin. Particular attention is paid to the case of primitive lattices in two-dimensions, where Rogers’ theorem fails. In this case, we apply a new theorem by Kleinbock and Yu to obtain a better error term than previous results due to Schmidt.

Superrigidity and arithmeticity for some aperiodic subsets in higher-rank simple Lie groups

Simon Machado, 24 March 2022

Abstract: Meyer sets are fascinating objects: they are aperiodic subsets of Euclidean spaces that nonetheless exhibit long-range aperiodic order. Sets of vertices of the Penrose tiling (P3) and Pisot-Vijarayaghavan numbers of a real number field are some of the most well-known examples. In his pioneering work, Meyer provided a powerful and elegant characterisation of Meyer sets. Years later, Lagarias proved a similar characterisation starting from what seemed to be considerably weaker assumptions. A fascinating question asks whether Meyer’s and Lagarias’ results may be extended to more general ambient groups. In fact, a first result in that direction was already obtained in Meyer’s work: he proved a sum-product phenomenon which, implicitly, boiled down to a classification of Meyer sets in the group of affine transformations of the line. I will talk about a generalisation of both Meyer’s and Lagarias’ theorems to discrete subsets of higher-rank simple Lie groups. I will explain how this result can be seen as a generalisation of Margulis’ arithmeticity theorem and how it can be deduced from Zimmer’s cocycle superrigidity. We will see that, surprisingly, Pisot-Vijarayaghavan numbers appear naturally in this context too.

Exact uniform approximation and Dirichlet spectrum
Johannes Schleischitz, 31 March 2022
Abstract: We consider the Dirichlet spectrum, with respect to maximum norm and simultaneous approximation. It is basically the analogue of the famous (multi-dimensional) Lagrange spectrum with respect to uniform approximation. By Dirichlet’s Theorem it is contained in [0,1]. The central new result is that it equals the entire interval [0,1] when the number of variables is two or more. We thereby get a new, constructive proof of a recent result by Beresnevich, Guan, Marnat, Ramirez and Velani that there are Dirichlet improvable vectors that are neither bad nor singular, in any dimension. We provide several generalizations, including metrical claims.
Video

Asymptotics of the equidistribution rate of expanding circles on compact hyperbolic quotients and applications
Emilio Corso, 7 April 2022
Abstract: Equidistribution properties of translates of orbits for subgroup actions on homogeneous spaces are intimately linked to the mixing features of the global action of the ambient group. The connection appears already in Margulis’ thesis (1969), displaying its full potential in the work of Eskin and McMullen during the nineties. On a quantitative level, the philosophy underpinning this linkage allows to transfer mixing rates to effective estimates for the rate of equidistribution, albeit at the cost of a sizeable loss in the exponent. In joint work with Ravotti, we instead resort to a spectral method, pioneered by Ratner in her study of quantitative mixing of geodesic and horocycle flows, in order to obtain the precise asymptotic behaviour of averages of regular observables along expanding circles on compact hyperbolic surfaces. The primary goal of the talk is to outline the salient traits of this method, illustrating how it leads to the relevant asymptotic expansion. In addition, we shall also present applications of the main result to distributional limit theorems and to quantitative error estimates on the corresponding hyperbolic lattice point counting problem, the latter having been examined, to date, only through number-theoretical methods in works of Selberg, Lax-Phillips and Phillips-Rudnick.
Video

Thin part of the arithmetic orbifolds
Mikolaj Fraczyk, 14 April 2022
Abstract: Let X be a symmetric space. The collar lemma, also known as the Margulis lemma, says that there exists an epsilon=epsilon(X), such that the epsilon-thin part of a locally symmetric space X/Γ looks locally like a quotient by a virtually unipotent subgroup. It turns out that in the arithmetic setting we can improve this lemma by making the epsilon grow linearly in the degree of the number filed generated by the traces of elements of Γ. I will explain why this is the case and present several applications, including the proof of the fact that an arithmetic locally symmetric manifold M is homotopy equivalent to a simplicial complex of size bounded linearly in the volume of M and degrees of all vertices bounded uniformly in terms of X. Based on a joint work with Sebastian Hurtado and Jean Raimbault.
Video

Effective Equidistribution on Hilbert Modular Surfaces
Ian Hoover, 28 April 2022
Subtitle: and an application to counting quadratic forms of square discriminant; Abstract: While ineffective equidistribution has been understood much more generally, effective results for non-compact orbits have been more scarce. I will give effective (polynomial) error rates for the translates of diagonal orbits on Hilbert modular surfaces. This work follows as a higher dimensional extension of the work of Kelmer and Kontorovich.
Video

Dynamical Borel–Cantelli Lemma for Lipschitz Twists

Jiajie Zheng, 5 May 2022

Abstract: In the study of some dynamical systems, the limit superior of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limit superior sets. However, the zero-one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this talk, we construct a generalized definition that can specialize into the shrinking targets and recurrence and our approach gives a unified proof to the zero-one laws for the two problems.

Fall 2021

Geometric Structures and Point Processes
Jayadev Athreya, 23 September 2021
Abstract: We’ll give several concrete examples of how to go from geometry to point processes, following work of Siegel, Veech, Masur, Eskin, Marklof, Mirzakhani, Wright, and others. We’ll discuss how this “probabilistic” perspective helps inform both the direction of questions one asks, as well as providing ideas of how to prove things. We’ll discuss some pieces of joint work with Cheung-Masur, Ghosh, Margulis, and Arana-Herrera.
Video

Height Gap, an Arithmetic Margulis Lemma and Almost Laws
Sebastian Hurtado, 30 September 2021
Abstract: We provide a new (more elementary) proof of a result of E. Breuillard, which state that a set of matrices with algebraic entries generating a non-virtually solvable group has a positive lower bound in its arithmetic height (we will explain this notion), this is a non-abelian version of Lehmer’s problem. We also show that in arithmetic locally symmetric spaces, short geodesics tend to be far from each other if the degree of the trace field is large. This lemma allows us to prove new results about growth of cohomology of sequences of locally symmetric spaces and to give a proof of a conjecture of Gelander. These results are works in progress with Joe Chen and Homin Lee, and with Mikolaj Fraczyk and Jean Raimbault.
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Khintchine’s theorem on manifolds
Lei Yang, 7 October 2021
Abstract: In this talk, we will prove the convergence part of Khitchine’s theorem on non-degenerate manifolds. This confirms a conjecture of Kleinbock and Margulis in 1998. Our approach uses geometric and dynamical ideas together with a new technique of `major and minor arcs’. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near `major arcs’ and give explicit exponentially small bounds for the measure of `minor arcs’. This is joint work with Victor Beresnevich.
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Equidistribution of degenerate curves and Dirichlet improvability

Pengyu Yang, 14 October 2021

Abstract: In the space of 3-lattices, we study the translates of a line segment under a diagonal flow. Sharp conditions for non-divergence and equidistribution will be given. As an application, we will show that Lebesgue-almost every point on a planar line is Dirichlet non-improvable if and only if the line is irrational. This is joint work with Kleinbock, de Saxcé and Shah. Generalizations to higher dimensions will also be discussed (work in progress with Shah).

The light cone Siegel transform, its moment formulas, and their applications

Dubi Kelmer, 21 October 2021

Abstract: In this talk I will describe an analogue of the Siegel transform where the role of Euclidean space is replaced by a light cone corresponding to an indefinite quadratic form.In this case one can use results on the spectral theory of incomplete Eisenstein series to establish moment formulas analogous to the classical formulas of Siegel, Rogers, and Schmidt.I will then describe several applications of these formulas to counting lattice points on the light cone, as well as for the distribution of rational points on the sphere. All new results are based on joint work with Shucheng Yu.

An inhomogeneous Khintchine-Groshev Theorem without monotonicity
Demi Allen, 28 October 2021
Abstract: The classical (inhomogeneous) Khintchine-Groshev Theorem tells us that for a monotonic approximating function $\psi: \mathbb{N} \to [0,\infty)$ the Lebesgue measure of the set of (inhomogeneously) $ψ$-well-approximable points in $\mathbb{R}^{nm}$ is zero or full depending on, respectively, the convergence or divergence of $\sum_{q=1}^{\infty}{q^{n-1}\psi(q)^m}$. In the homogeneous case, it is now known that the monotonicity condition on $\psi$ can be removed whenever $nm>1$ and cannot be removed when $nm=1$. In this talk I will discuss recent work with Felipe A. Ramírez (Wesleyan, US) in which we show that the inhomogeneous Khintchine-Groshev Theorem is true without the monotonicity assumption on $\psi$ whenever $nm>2$. This result brings the inhomogeneous theory almost in line with the completed homogeneous theory. I will survey previous results towards removing monotonicity from the homogeneous and inhomogeneous Khintchine-Groshev Theorem before discussing the main ideas behind the proof our recent result.
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Quantitative equidistribution and Randomness
Alexander Gorodnik, 4 November 2021
Abstract: We discuss some results on quantitative equidistribution on homogeneous spaces and related problems about behaviour of arithmetic counting functions. This is a joint work with Björklund and Fregoli.
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Pair correlation of monomial sequences modulo 1
Chris Lutsko, 11 November 2021
Abstract: Fix $\alpha, \theta > 0$, and consider the sequence $(\alpha n^\theta \mod 1)_{n>0}$. Since the seminal work of Rudnick-Sarnak (1998), and due to the Berry-Tabor conjecture in quantum chaos, the fine-scale properties of these dilated mononomial sequences have been intensively studied. In this talk, I will briefly survey what is known about these sequences and present a recent result (joint with Sourmelidis and Technau) showing that for $\theta \le 1/3$, and $\alpha > 0$, the pair correlation function is Poissonian. While the techniques we use are derived from analytic number theory, the problem is rooted in dynamics and relates to dynamical proofs for related problems.
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On a theorem of Davenport-Schmidt on Dirichlet improvable pairs

Anurag Rao, 18 November 2021

Abstract: In 1970, Davenport and Schmidt studied a Diophantine property of pairs of real numbers; it concerned those pairs for which the classical Dirichlet theorem can be improved. They showed that the set of Dirichlet-improvable pairs, while small in the sense of having zero Lebesgue measure, has full Hausdorff dimension. We study a similar Dirichlet-improvable property, where the approximations are made using an arbitrary norm rather than the supremum norm, and show the same result. To this end this, we recast the Dirichlet-improvable property into a dynamical property of certain orbits in the space of unimodular lattices, and prove a Hajos-Minkowski type result in the geometry of numbers. This is joint work with Dmitry Kleinbock.

Two-step equidistribution for bi-quadratic torus packets

Ilya Khayutin, 2 December 2021

Abstract: A major challenge to the asymptotic analysis of a sequence of probability measures on a homogeneous space, invariant under diagonalizable groups, is the possibility of accumulation on intermediate homogeneous subspaces. In this aspect higher rank homogeneous flows cannot be expected to share the rigidity properties of unipotent ones. In particular, the linearization technique fails for diagonalizable flows. In a joint work in progress with A. Wieser we show how in favorable situations one can actually use the existence of intermediate homogeneous spaces in our benefit. We show that periodic measures on some packets of periodic torus orbits on PGL4(Z)\PGL4(R) converge in the limit to a measure with a non-trivial Haar component. The proof goes by establishing high entropy for the limit measure. The method utilizes the intermediate homogeneous space to split the analysis into two more tractable steps.

On the image in the torus of sparse points on expanding analytic curves

Michael Bersudsky, 9 December 2021

Abstract: It is known that the projection to the 2-torus of the normalised parameter measure on a circle of radius $R$ in the plane becomes uniformly distributed as $R$ grows to infinity. I will discuss the following natural discrete analogue for this problem. Starting from an angle and a sequence of radii {$R_n$} which diverges to infinity, I will consider the projection to the 2-torus of the n’th roots of unity rotated by this angle and dilated by a factor of $R_n$. The interesting regime in this problem is when $R_n$ is much larger than n so that the dilated roots of unity appear sparsely on the dilated circle.I will discuss 3 types of results: 1) Validity of equidistribution for all angles when the sparsity is polynomial; 2) Failure of equidistribution for some super polynomial dilations; 3) Equidistribution for almost all angles for arbitrary dilations. I will discuss the above type of results in greater generality and I will try to explain how the theory of o-minimal structures is related to the proof.

Spring 2021

Expanding measures and random walks on homogeneous spaces
Cagri Sert, 1 February 2021
Abstract: We will start by reviewing some recent works on random walks on homogeneous spaces. We will continue by discussing the notion of a H-expanding probability measure on a connected semisimple Lie group H, that we introduce inspired by these developments. As we shall see, for a H-expanding µ with H < G, on the one hand, one can obtain a description of µ-stationary probability measures on the homogeneous space G/Λ using the measure classification results of Eskin– Lindenstrauss, and on the other hand, the recurrence techniques of Benoist–Quint can be generalized to this setting. As a result, we will deduce equidistribution and orbit closure description results simultaneously for a class of subgroups which contains Zariski-dense subgroups and some epimorphic subgroups of H. If time allows, we will see how, using an idea of Simmons–Weiss, this allows also us to deduce Birkhoff genericity of a class of fractal measures with respect to expanding diagonal actions. Joint work with Roland Prohaska and Ronggang Shi.
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Classification and statistics of cut-and-project sets
Barak Weiss, 8 February 2021
Abstract: We introduce a class of so-called “Ratner-Marklof-Strombergsson measures”. These are probability measures supported on cut-and-project sets in Euclidean space of dimension d>1 which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R) (affine or linear maps preserving orientation and volume). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project sets. Joint work with Rene Ruehr and Yotam Smilansky.
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Equidistribution of affine random walks on some nilmanifolds

Tsviqa Lakrec, 22 February 2021

Abstract: We consider the action of the group of affine transformations on a nilmanifold. Given a probability measure on this group and a starting point, a random walk on the nilmanifold is defined. We study quantitative equidistribution in law of such affine random walks on nilmanifolds. Under certain assumptions, we show that a failure to have fast equidistribution on a nilmanifold is due to a failure on some factor nilmanifold. Combined with equidistribution results on the torus, this leads to an equidistribution statement on some nilmanifolds, such as Heisenberg nilmanifolds. This talk is based on joint works with Weikun He and Elon Lindenstrauss.

Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends
Minju Lee, 8 March 2021
Abstract: This is joint work with Hee Oh. We establish an analogue of Ratner’s orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\mathrm{SO}(d,1)$ acting on the space $\Gamma\backslash\mathrm{SO}(d,1)$, assuming that the associated hyperbolic manifold $M=\Gamma\backslash\mathbb{H}^d$ is a convex cocompact manifold with Fuchsian ends. For $d = 3$, this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate subgroups causes serious issues, but in the end, all orbit closures of unipotent flows are relatively homogeneous. Our results imply the following: for any $k\geq 1$, (1) the closure of any $k$-horosphere in $M$ is a properly immersed submanifold; (2) the closure of any geodesic $(k+1)$-plane in $M$ is a properly immersed submanifold; (3) an infinite sequence of maximal properly immersed geodesic $(k+1)$-planes intersecting $\mathrm{core} M$ becomes dense in $M$.
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Towards an extreme value law for the deepest cusp excursions of the unipotent flow
Maxim Kirsebom, 15 March 2021
Abstract: The unipotent flow on the unit tangent bundle of the modular surface is a classic example of a homogeneous flow when understood through the identification with PSL₂(R)/PSL₂(Z). The ergodicity of the flow implies that almost every orbit is dense in the space and hence must eventually make excursions deeper and deeper into the cusp. We are interested in understanding the nature of these excursions. In the described setting, and more generally, Athreya and Margulis proved that the maximal excursions obey the logarithm law almost surely, meaning that their growth rate scales the logarithm of the time. In this work we focus on a more precise description of this behaviour, namely determining the probability that the deepest excursion fails to outperform the expected asymptotic behaviour by an additive amount. This question may be phrased in the language of extreme value statistics and we establish some results towards a complete extreme value law in this setting. The methods used are based on classical ideas from geometry of numbers. This is work in progress, joint with Keivan Mallahi-Karai.
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On the dimension of self-similar measures
Peter Varju, 22 March 2021
Abstract: Let f1,…,fn be a collection of contracting similarities on R, and let p1,…,pn be a probability vector. There is a unique probability measure mu on R that satisfies the identity mu = p1 f1(mu) + … + pn fn(mu). This measure is called self-similar. The maps f1,…,fn are said to satisfy the no exact overlaps condition if they generate a free semigroup (i.e. all compositions are distinct). Under this condition, the dimension of mu is conjectured to be the minimum of 1 and the ratio of the entropy of p1,…,pn and the average logarithmic contraction factor of the fi. This conjecture has been recently established in some special cases, including when n=2 and f1 and f2 have the same contraction factor. In the talk I will discuss recent progress by Ariel Rapaport and myself in the case n=3. In this case new difficulties arise as was demonstrated by recent examples of Baker and Barany, Kaenmaki of IFS’s with arbitrarily weak separation properties.
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Minimal vectors in $\C^2$ and best constant for Dirichlet theorem over $\C$
Nicolas Chevallier, 5 April 2021
Abstract: We study minimal vectors in lattices over Gaussian integers in $\C^2$.We show that the index of the sub-lattice generated by two consecutive minimal vectors in a lattice of $\C^2$, can be either $1$ or $2$.Next, we describe the constraints on pairs of consecutive minimal vectors. These constraints make it possible to find the best constant for Dirichlet theorem about approximations of complex numbers by quotient of Gaussian integers.
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Rational numbers near self-similar sets
Han Yu, 12 April 2021
Abstract: We will discuss a problem on counting rational numbers near self-similar sets. In particular, we will show that the set of rational numbers is ‘reasonably well distributed’ around the middle $p$-th Cantor set when $p$ is a large integer. Our approach is via Fourier analysis and we will also discuss some problems on Fourier transform of self-similar measures which are of independent interest. As a result, it is possible to show that $p=5$ satisfies the previous statement. The materials come from various working-in-progress projects with D. Allen, S. Chow and P. Varju.
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Using templates to study problems in dynamics and number theory
Tushar Das, 19 April 2021
Abstract: Templates may be viewed as a combinatorial device that helps study asymptotic properties of lattice successive minima. This simple idea, introduced in joint work with Lior Fishman, David Simmons, and Mariusz Urbanski, promises to be useful in several areas beyond our current applications. The latter lie at the fertile interface along Dani’s correspondence principle between Diophantine approximation and homogeneous flows, deepened by Kleinbock & Margulis; and Schmidt & Summerer’s parametric geometry of numbers, deepened by Roy. Templates are at the heart of our variational principle (arXiv:1901.06602), which provides a unified framework to compute the Hausdorff and packing dimensions of a variety of sets of dynamical and number-theoretic interest. We will introduce and give some flavor for our project, hint at a few new directions, and hope to present several open problems of varying depth to reward participants of this wonderful seminar!
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An application of Margulis’ inequality to effective equidistribution

Asaf Katz, 26 April 2021

Abstract: Ratner’s celebrated equidistribution theorem states that the trajectory of any point in a homogeneous space under a unipotent flow is getting equidistributed with respect to some algebraic measure. In the case where the action is horospherical, one can deduce an effective equidistribution result by mixing methods, an idea that goes back to Margulis’ thesis. When the homogeneous space is non-compact, one needs to impose further “diophantine conditions” over the base point, quantifying some recurrence rates, in order to get a quantified equidistribution result. In the talk I will discuss certain diophantine conditions, and in particular I will show how a new Margulis’ type inequality for translates of horospherical orbits helps verify such conditions. This results in a quantified equidistribution result for a large class of points, akin to the results of A. Strombreggson dealing with SL2 case. In particular we deduce a fully effective quantitative equidistribution for horospherical trajectories of lattices defined over number fields, without pertaining to the strong subspace theorem.

Generalization of Selberg’s 3⁄16 theorem for convex cocompact thin subgroups of SO(n, 1)

Pratyush Sarkar, 3 May 2021

Abstract: Selberg’s 3/16 theorem for congruence covers of the modular surface is a beautiful theorem which has a natural dynamical interpretation as uniform exponential mixing. Bourgain-Gamburd-Sarnak’s breakthrough works initiated many recent developments to generalize Selberg’s theorem for infinite volume hyperbolic manifolds. One such result is by Oh-Winter establishing uniform exponential mixing for convex cocompact hyperbolic surfaces. These are not only interesting in and of itself but can also be used for a wide range of applications including uniform resonance free regions for the resolvent of the Laplacian, affine sieve, and prime geodesic theorems. I will present a further generalization to higher dimensions and some of these immediate consequences.

Counting problems on a random lattice
Seungki Kim, 10 May 2021
Abstract: A random lattice is a random element of SL(n,Z) \ SL(n,R) equipped with the probability measure inherited from the Haar measure of SL(n,R). Analogous to the usual lattice point-counting, one tries to “count” — more precisely, study the statistics of — the random lattice points inside a ball or other shapes. I’ll give a gentle introduction to this topic, discussing the early works of Siegel, Rogers and Schmidt and some of the recent results, as well as their applications.
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Fall 2020

A subspace theorem for manifolds

Emmanuel Breuillard, 18 September 2020

Abstract: Schmidt’s subspace theorem is a fundamental result in diophantine approximation and a natural generalization of Roth’s celebrated theorem. In this talk I will discuss a geometric understanding of this theorem that blends homogeneous dynamics and geometric invariant theory. Combined with the Kleinbock-Margulis quantitative non-divergence estimates this yields a natural generalization of the subspace theorem to systems of linear forms that depend nicely on a parameter. I will also present several applications and consequences of the main result. Joint work with Nicolas de Saxcé.

Multiscale substitution tilings

Yotam Smilansky, 25 September 2020

Abstract: Multiscale substitution tilings are a new family of tilings of Euclidean space that are generated by multiscale substitution rules. Unlike the standard setup of substitution tilings, which is a basic object of study within the aperiodic order community and includes examples such as the Penrose and the pinwheel tilings, multiple distinct scaling constants are allowed, and the defining process of inflation and subdivision is a continuous one. Under a certain irrationality assumption on the scaling constants, this construction gives rise to a new class of tilings, tiling spaces and tiling dynamical systems, which are intrinsically different from those that arise in the standard setup. In the talk I will describe these new objects and discuss various structural, geometrical, statistical and dynamical results. Based on joint work with Yaar Solomon.

Counting social interactions for discrete subsets of the plane
Samantha Fairchild, 2 October 2020
Abstract: Given a discrete subset V in the plane, how many points would you expect there to be in a ball of radius 100? What if the radius is 10,000? Due to the results of Fairchild and forthcoming work with Burrin, when V arises as orbits of non-uniform lattice subgroups of SL(2,R), we can understand asymptotic growth rate with error terms of the number of points in V for a broad family of sets. A crucial aspect of these arguments and similar arguments is understanding how to count pairs of saddle connections with certain properties determining the interactions between them, like having a fixed determinant or having another point in V nearby. We will focus on a concrete case used to state the theorem and highlight the proof strategy. We will also discuss some ongoing work and ideas which advertise the generality and strength of this argument.
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Effective equidistribution of horospherical flows in infinite volume

Nattalie Tamam, 9 October 2020

Abstract: We want to provide effective information about averages of orbits of the horospherical subgroup acting on a hyperbolic manifold of infinite volume. We start by presenting the setting and results for manifolds with finite volume. Then, discuss the difficulties that arise when studying the infinite volume setting, and the measures that play a crucial role in it. This is joint work with Jacqueline Warren.

Decimation Limits of Algebraic Actions
Douglas Lind, 16 October 2020
Abstract: This is intended to be an expository talk using simple examples to illustrate what’s going on, and so will (hopefully) be a gentle introduction to these topics. Given a polynomial in d commuting variables we can define an algebraic action of ℤ^d by commuting automorphisms of a compact subgroup of 𝕋^{(ℤ^d)}. Restricting the coordinates of points in this group to finite-index subgroups of ℤ^d gives other algebraic actions, defined by polynomials whose support grows polynomially and whose coefficients grow exponentially. But by “renormalizing” we can obtain a limiting object that is a concave function on ℝ^d with interesting properties, e.g. its maximum value is the entropy of the action. For some polynomials this function also arises in statistical mechanics models as the “surface tension” of a random surface via a variational principle. In joint work with Arzhakova, Schmidt, and Verbitskiy, we establish this limiting behavior, and identify the limit in terms of the Legendre transform of the Ronkin function of the polynomial. The proof is based on Mahler’s estimates on polynomial coefficients using Mahler measure, and an idea used by Boyd to prove that Mahler measure is continuous in the coefficients of the polynomial. Refinements of convergence questions involve diophantine issues that I will discuss, together with some open problems.
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Small values at integer points of generic subhomogeneous functions

Mishel Skenderi, 23 October 2020

Abstract: This talk will be based on joint work with Dmitry Kleinbock that has been motivated by several recent papers (among them, those of Athreya-Margulis, Bourgain, Ghosh-Gorodnik-Nevo, Kelmer-Yu). Given a certain sort of group $G$ and certain sorts of functions $f: \mathbb{R}^n \to \mathbb{R}$ and $ψ : \mathbb{R}ⁿ → \mathbb{R}_>0,$ we obtain necessary and sufficient conditions so that for Haar-almost every $g ∈ G,$ there exist infinitely many (respectively, finitely many) $v \in \mathbb{Z}^n$ for which $ \lvert (f ˆ g)(v) \rvert ≤ ψ(\lVert v\rVert),$ where $\lVert\cdot\rVert$ is an arbitrary norm on $\mathbb{R}ⁿ.$ We also give a sufficient condition in the setting of uniform approximation. As a consequence of our methods, we obtain generalizations to the case of vector-valued (simultaneous) approximation with no additional effort. In our work, we use probabilistic results in the geometry of numbers that go back several decades to the work of Siegel, Rogers, and W. Schmidt; these results have recently found new life thanks to a 2009 paper of Athreya-Margulis.

Intrinsic Diophantine Approximation of circles
Byungchul Cha, 6 November 2020
Abstract: Let $S^1$ be the unit circle in $\mathbb{R}^2$ centered at the origin and let $Z$ be a countable dense subset of $S^1$, for instance, the set $Z = S^1(\mathbb{Q})$ of all rational points in $S^1$. We give a complete description of an initial discrete part of the Lagrange spectrum of $S^1$ in the sense of intrinsic Diophantine approximation. This is an analogue of the classical result of Markoff in 1879, where he characterized the most badly approximable real numbers via the periods of their continued fraction expansions. Additionally, we present similar results for a few different subsets $Z$ of $S^1$. This is joint work with Dong Han Kim.
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Joining classification and factor rigidity in infinite volume

Jacqueline Warren, 13 November 2020

Abstract: For a group acting on two spaces, a joining of these systems is a measure on the product space that is invariant under the diagonal action and projects to the original measures on each space. Joinings are a powerful tool in ergodic theory, and joinings for the horocycle flow were classified by Ratner in the finite volume setting, with many interesting applications. In this talk, I will discuss some of these applications and present joining classification for horospherical flows in the infinite volume setting, as well as a key factor rigidity theorem that is used in the proof. This talk is intended to be accessible to graduate students.

On the dimension drop conjecture for diagonal flows on the space of lattices

Shahriar Mirzadeh, 20 November 2020

Abstract: Consider the set of points in a homogeneous space X=G/Gamma whose g_t orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when it has real rank 1. In this talk we will prove the conjecture for probably the most important example of the higher rank case, namely: G=SL(m+n, R), Gamma=SL(m+n,Z), and g_t = diag(exp(t/m), …, exp(t/m), exp(-t/n), …, exp(-t/n)). We can also use our main result to produce new applications to Diophantine approximation. This project is joint work with Dmitry Kleinbock.

Large centralizers and counting integral points on affine varieties

Osama Khalil, 4 December 2020

Abstract: Duke-Rudnick-Sarnak and Eskin-McMullen initiated the use of ergodic methods to count integral points on affine homogeneous varieties. They reduced the problem to one of studying limiting distributions of translates of periods of reductive groups on homogeneous spaces. The breakthrough of Eskin, Mozes and Shah provided a rather complete understanding of this question in the case the reductive group has a “small centralizer” inside the ambient group. In this talk, we describe work in progress giving new results on the equidistribution of generic translates of closed orbits of semisimple groups with “large centralizers”. The key new ingredient is an algebraic description of a partial compactification (for lack of a better word) of the set of intermediate groups which act as obstructions to equidistribution. This allows us to employ tools from geometric invariant theory to study the avoidance problem.

Gaps of saddle connection directions for some branched covers of tori
Anthony Sanchez, 11 December 2020
Abstract: Holonomy vectors of translation surfaces provide a geometric generalization for higher genus surfaces of (primitive) integer lattice points. The counting and distribution properties of holonomy vectors on translation surfaces have been studied extensively. In this talk, we consider the following question: How random are the holonomy vectors of a translation surface? We motivate the gap distribution of slopes of holonomy vectors as a measure of randomness and compute the gap distribution for the class of translation surfaces given by gluing two identical tori along a slit. No prior background on translation surfaces or gap distributions will be assumed.
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