Counting Matrices Over Finite Rank Multiplicative Groups

2:00pm, Wednesday 11 June

Abstract

There have been many recent works regarding arithmetic statistics questions related to matrices with entries from sets of number theoretic interest. This includes, in particular, providing upper bounds on the number of matrices with a prescribed rank, determinant, or characteristic polynomial, over such a set. Motivated by some recent work by Alon and Solymosi (2023), we consider matrices with entries from finitely generated subgroups of the group of units of a field of characteristic zero. Such sets require a considerably different approach to many that have been studied previously. The primary tools we require follow from the Subspace Theorem of Schmidt (1989) on the simultaneous approximation of algebraic numbers by rational numbers.