On commuting pairs in arbitrary sets of 2x2 matrices

TL;DR

This paper investigates the structure of pairs of commuting matrices in arbitrary sets of 2x2 real matrices, establishing conditions under which such pairs are rare or concentrated in low-dimensional subspaces, with implications for incidence geometry and additive combinatorics.

Contribution

The authors provide a quantitative characterization of commuting pairs in matrix sets, connecting algebraic, geometric, and additive combinatorics techniques, and extend results to structured measures on real numbers.

Findings

Either the probability of commuting pairs is very small or matrices are concentrated in a low-dimensional subspace.

Established bounds for measures supported on generalized arithmetic or multiplicative progressions.

Connected the problem to incidence geometry, growth in groups, and sum-product estimates over the reals.

Abstract

Let $\textrm{Mat}_2(\mathbb{R})$ be the set of $2 \times 2$ matrices with real entries. For any $\varepsilon>0$ and any finitely--supported probability measure $\mu$ on $\textrm{Mat}_2(\mathbb{R})$, we prove that either \[ T(\mu) = \sum_{X, Y \in {\rm supp}(\mu), XY = YX} \mu(X) \mu(Y) < \varepsilon \] or there exists some finite set ${S}$ contained in a $2$-dimensional subspace of $\textrm{Mat}_2(\mathbb{R})$ such that $\mu({S}) \geq \varepsilon/8$. This is sharp up to the multiplicative constant. We prove quantitatively stronger results when \[ \mu ( (a_{i,j})_{1 \leq i,j \leq 2} ) = \nu(a_{1,1}) \dots \nu(a_{2,2}) \ \ \text{for every} \ a_{1,1}, \dots, a_{2,2} \in \mathbb{R}, \] with $\nu$ being some finitely--supported probability measure on $\mathbb{R}$. For instance, when ${A} \subset \mathbb{R}$ is a generalised arithmetic progression or multiplicative progression of dimension $d$ and $\nu = {1}_{{A}}/|{A}|$, our techniques imply that $|{A}|^{-3} \ll_d T(\mu) \ll_d |{A}|^{-3}$. Our methods highlight the connections of this problem to results in incidence geometry, growth in groups phenomenon as well as Bourgain--Chang type sum-product estimates over $\mathbb{R}$. The latter includes applications of Schmidt's subspace theorem and the resolution of the weak polynomial Freiman--Ruzsa conjecture over integers.