Counting commuting integer matrices

Abstract: Consider the set of pairs of $d\times d$ matrices $(A,B)$ whose entries are all integers with absolute value at most $N$. We call $(A,B)$ a \emph{commuting pair} if $AB=BA$. Browning, Sawin, and Wang recently showed that the number of commuting pairs is at most $O_d(N^{d^2 + 2 - \frac{2}{d +1}})$. They further conjectured that the lower bound $\Omega_d(N^{d^2 + 1})$, which comes from letting $A$ or $B$ be a multiple of the identity matrix, should be sharp. In this talk, I will discuss progress on the cases $d=2$ and $d=3$, where we show that this conjecture holds. I will also demonstrate how our approach relates counting commuting pairs of matrices to the study of restricted divisor correlations in number theory.\\ Joint work with Akshat Mudgal (University of Warwick)

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)