On commuting pairs in arbitrary sets of 2x2 matrices

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.