On commuting integer matrices

TL;DR

This work counts commuting pairs of $d\times d$ integer matrices with entries in $[-N,N]$, focusing on $d\in\{2,3\}$. The authors establish sharp bounds and an explicit asymptotic: for $d=3$, $N^{10} \ll \mathfrak{C}_3(N) \ll N^{10}$, confirming a prior conjecture, while for $d=2$, they prove an asymptotic formula $\mathfrak{C}_2(N) = K(2N+1)^5(1+o(1))$ with an explicit constant $K = \frac{10\zeta(2)}{3\zeta(3)}$. The methods are elementary and rely on precise restricted divisor correlations, plus a careful case analysis of the linear constraints arising from AB=BA, including a p-adic local analysis for $d=2$. The results illuminate the structure of commuting matrices and the role of degenerate solutions, and they align with the conjectured order of magnitude while providing an exact main term for $d=2$ and a sharp upper bound for $d=3$, with insights into local densities and potential circle-method limitations in this setting.

Abstract

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

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1.4
• Lemma 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof