# Product sets of arithmetic progressions

## Location

384H

Thursday, February 10, 2022 2:00 PM

Max Wenqiang Xu

We prove that the size of the product set of any finite arithmetic progression A in integers with size N is at least N^{2}/(log N)^{c+o(1)}, where 2c=1-(1+loglog 2)/(log 2). This matches the bound in the celebrated Erd*ő*s multiplication table problem, up to a factor of (log N)^{o(1)} and thus confirms a conjecture of Elekes and Ruzsa from about two decades ago.

If instead A is relaxed to be a subset of a finite arithmetic progression in integers with positive constant density, we prove that the product set is at least N^{2}/(log N)^{2log 2 - 1 + o(1)}. This solves the typical case of another conjecture of Elekes and Ruzsa on the size of the product set of a set A whose sum set is of size O(|A|).

This is joint work with Yunkun Zhou.