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 N^10 C₃ (N) N^10, 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.
Chapman et al. (Tue,) studied this question.