Winning property of counterexamples to Uniform Littlewood’s Conjecture

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.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar