SOME COUNTING QUESTIONS FOR MATRIX PRODUCTS

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 $A_1 \cdots A_m$ , 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 and solution sets of related equations such as $A_1 \cdots A_m=C$ and $A_1 \cdots A_m=B_1 \cdots B_m$ . 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.