On commuting integer matrices

Given d, N ∈ N, we define Cd(N) to be the number of pairs of d× d matrices $A,B$ with entries in [-N,N] ∩ Z such that $AB = BA$. We prove that N¹⁰ C₃(N) N¹⁰, thus confirming a speculation of Browning-Sawin-Wang. We further establish that C₂(N) = K(2N+1)⁵ (1 + o(1)), where $K>0$ is an explicit constant. Our methods are completely elementary and rely on upper bounds of the correct order for restricted divisor correlations with high uniformity.