On commuting pairs in arbitrary sets of 2x2 matrices

TL;DR

This work analyzes the number of commuting pairs arising from random $2\times2$ real matrices by studying a probability measure $\mu$ on $\mathrm{Mat}_2(\mathbb{R})$ and the associated commutativity energy $T(\mu)$. A sharp dichotomy is established: either $T(\mu)$ is small, or a substantial portion of the mass concentrates on a 2-dimensional subspace, quantified by $\delta(\mu)$; this hinges on a precise structural understanding of when two random matrices commute. For product-type measures $\mu_{\nu}$, the authors derive quantitative bounds in terms of typical energy quantities $M(\nu)$ and $\|\nu\|_p$, linking the problem to sum-product phenomena, incidence geometry, and growth in groups, and they connect these results to Schmidt’s subspace theorem and the weak polynomial Freiman–Ruzsa conjecture. The paper then develops parallel bounds for sets $\mathcal{A} \subset \mathbb{R}$ with small sum/product structure, including both additive and multiplicative doubling regimes, and extends the analysis to affine transformations to bound $T(\mu_{\nu})$ in terms of the additive/multiplicative structure of $\mathrm{supp}(\nu)$, culminating in a robust set of results that illuminate the interplay between commutativity, structure, and energy in low-dimensional matrix groups.

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 $μ$ on $\textrm{Mat}_2(\mathbb{R})$, we prove that either \[ T(μ) = \sum_{X, Y \in {\rm supp}(μ), XY = YX} μ(X) μ(Y) < \varepsilon \] or there exists some finite set ${S}$ contained in a $2$-dimensional subspace of $\textrm{Mat}_2(\mathbb{R})$ such that $μ({S}) \geq \varepsilon/8$. This is sharp up to the multiplicative constant. We prove quantitatively stronger results when \[ μ( (a_{i,j})_{1 \leq i,j \leq 2} ) = ν(a_{1,1}) \dots ν(a_{2,2}) \ \ \text{for every} \ a_{1,1}, \dots, a_{2,2} \in \mathbb{R}, \] with $ν$ 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 $ν= {1}_{A}/|{A}|$, our techniques imply that $|{A}|^{-3} \ll_d T(μ) \ll_d |{A}|^{-3}$. 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.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Proposition 1.5
• Corollary 1.6
• Theorem 1.7
• Conjecture 1.8
• Lemma 2.1
• Lemma 2.2