New England Dynamics and Number Theory Seminar
Fall 2026
| A zero–one law for twisted Diophantine approximation |
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| Taehyeong Kim, Brandeis University, 8 October 2026 |
| Abstract: We study twisted Diophantine approximation, where the matrix is fixed and the target varies. I will present an exact Lebesgue measure zero–one law for arbitrary matrices and non-increasing approximation functions satisfying dyadic regularity. The criterion corrects the usual volume sum by homogeneous lattice-point counts and can be expressed through all successive minima of an associated diagonal lattice trajectory, capturing clustering in every direction. I will also discuss applications to inhomogeneous approximation constants and Hausdorff dimension. This talk is based on a joint project with Vasiliy Neckrasov; the other part establishes Hausdorff dimension formulae in uniform twisted Diophantine approximation. |
| Video |
| An introduction to uniform multiplicative Diophantine approximation |
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| Dmitry Kleinbock, Brandeis University, 22 October 2026 |
| Abstract: The main purpose of this talk is to give a gentle introduction to Chengyang's lecture that will start right after I finish. I'll give a motivation for the uniform version of Littlewood’s Conjecture (stronger than the original one), and explain how to prove, using ergodic theory on the space of lattices, that its set of exceptions has measure zero. Joint work with Prasuna Bandi and Reynold Fregoli. |
| Video |
| Winning property of counterexamples to Uniform Littlewood’s Conjecture |
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| Chengyang Wu, University of Chicago, 22 October 2026 |
| Abstract: Littlewood’s conjecture is a long-standing open problem in multiplicative Diophantine approximation. Its uniform version was recently disproved by Schleischitz, as expected by Bandi, Fregoli, and Kleinbock in 2025. A natural follow-up question concerns the Hausdorff dimension of the set of its counterexamples. In this talk, we present a new approach to studying the structure of this set and establish a winning property for it. The key ingredient is an adaptation of An’s simplex lemma (2016). In particular, our result implies that, for any two-dimensional weight, the intersection of the counterexample set with the corresponding set of weighted badly approximable vectors has full Hausdorff dimension. This is a joint work with Vasiliy Neckrasov and Bohan Yang. |
| Video |
| Winning property of counterexamples to Uniform Littlewood’s Conjecture |
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| Chengyang Wu, University of Chicago, 22 October 2026 |
| Abstract: Littlewood’s conjecture is a long-standing open problem in multiplicative Diophantine approximation. Its uniform version was recently disproved by Schleischitz, as expected by Bandi, Fregoli, and Kleinbock in 2025. A natural follow-up question concerns the Hausdorff dimension of the set of its counterexamples. In this talk, we present a new approach to studying the structure of this set and establish a winning property for it. The key ingredient is an adaptation of An’s simplex lemma (2016). In particular, our result implies that, for any two-dimensional weight, the intersection of the counterexample set with the corresponding set of weighted badly approximable vectors has full Hausdorff dimension. This is a joint work with Vasiliy Neckrasov and Bohan Yang. |
| Video |
| Limit theorems for cuspidal orbits on Fuchsian groups |
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| Jungwon Lee, Caltech, 12 November 2026 |
| Abstract: We discuss the dynamics of Bowen-Series maps, which generalise classical continued fraction maps to the setting of general Fuchsian groups and provide symbolic codings for geodesic flows on hyperbolic surfaces. We present a transfer operator framework to study statistical properties of their cuspidal orbits, and in turn, establish limit laws for quantum modular forms at rational points of bounded denominator. Examples include classical modular symbols and central values of additive twists of automorphic L-functions (joint with Sandro Bettin and Sary Drappeau). |