On commuting integer matrices

Source abstract

Given d , N ∈ N d, N \in \mathbb {N} , we define C d ( N ) \mathfrak {C}_d(N) to be the number of pairs of d × d d\times d matrices A , B A,B with entries in [ − N , N ] ∩ Z [-N,N] \cap \mathbb {Z} such that A B = B A AB = BA . We prove that $$N 10 ≪ C 3 ( N ) ≪ N 10 , N^{10} \ll \mathfrak {C}_3(N) \ll N^{10},$$ thus confirming a speculation of Browning–Sawin–Wang. We further establish that $$C 2 ( N ) = K ( 2 N + 1 ) 5 ( 1 + o ( 1 ) ) , \mathfrak {C}_2(N) = K(2N+1)^5 (1 + o(1)),$$ where K > 0 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.