A variation on the p-adic Littlewood Conjecture

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 viewpoints: the Diophantine-approximation one and the dynamical one. Using the dynamical viewpoint, we rephrase the conjecture using Hecke neighbors and prove a partial statement towards the conjecture. Namely, we prove that the Hausdorff dimension of the exception set is strictly smaller than 1. Our tools for the proof are mainly the effective equidistribution of Hecke neighbors due to Oh et al and to expander properties of $SL_2(Z/pZ)$ due to Bourgain-Gamburd. This talk is based on an ongoing joint work with Erez Nesharim.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar