Some counting questions for matrix products

Abstract

Given a set $X$ of $n\times n$ matrices and a positive integer $m$, we consider the problem of estimating the cardinalities of the product sets $A_1 \dotsc A_m$, where $A_i\in X$. When $X=\mathcal M_n(\mathbb{Z};H)$, the set of $n\times n$ matrices with integer elements of size at most $H$, we give several bounds on the cardinalities of the product sets. Related to this result, we also give some bounds on the cardinalities of the set of solutions of the related equations such as $A_1 \dotsc A_m=C$ and $A_1 \dotsc A_m=B_1 \dotsc B_m$. We also consider the case where $X$ is the subset of matrices in $\mathcal M_n(\mathbb{F})$, where $\mathbb{F}$ is a field, with bounded rank $k\leq n$. In this case, we completely classify the related product set.