Modul: MAT760 Ergodic theory and dynamical systems seminar

## Non Algebraic Versions of the p-Adic Littlewood Conjecture and of Duke’s Theorem for Subcollections

Talk by Yuval Yifrach

Speaker invited by: Prof. Dr. Alexander Gorodnik

**Date:** 06.11.23 **Time:** 15.00 - 16.00 **Room:** Y27H25

Various algebraic phenomenons in homogeneous dynamics have non algebraic counterparts. For example, the equidistribution of Hecke neighbors can be seen as a non-algebraic counterpart of Duke’s Theorem. In this talk, we consider non-algebraic counterparts of the p-Adic Littlewood Conjecture and of Duke’s Theorem for subcollections.

One of the non-algebraic counterparts of the p-Adic Littlewood Conjecture involves unboundedness of the A-orbits of arbitrary choices of p-Hecke neighbors of a lattice as p goes to infinity along the primes.

We prove, using expanders, a bootstrap argument and the equidistribution of Hecke neighbors, that the set of exceptions for this conjecture has Hausdorff dimension strictly smaller than 1 in [0,1] (where we assign lattices to points in [0,1]). Moreover, we discuss evidence for the conjecture in some cases using GRH.

This talk is based on a joint ongoing work with Erez Nesharim from the Technion.