On commuting pairs in arbitrary sets of 2x2 matrices

Abstract

Let ${\text{Mat}}_2(\mathbb{R})$ be the set of $2\times 2$ matrices with real entries. For any $\epsilon >0$ and any finitely--supported probability measure $\mu$ on ${\text{Mat}}_2(\mathbb{R})$, 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 ${\text{Mat}}_2(\mathbb{R})$ such that $\mu (S)\ge \epsilon /8$. 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 $\nu$ 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.

arXiv comment23 pages, Significant revisions: Theorem 1.1 strengthened, Added connection to growth in groups, see Theorem 1.7, Conjecture 1.8 and newly added section 7, and updated references