Let \(\mathbb {F}\) be a field and let \(\text {Mat}_2(\mathbb {F})\) be the set of \(2 \times 2\) matrices with entries from \(\mathbb {F}\) . For any \(\varepsilon >0\) and any finitely–supported probability measure \(\mu \) on \(\text {Mat}_2(\mathbb {F})\) , we prove that either \(\begin{aligned} T(\mu ) = \sum _{X, Y \in \textrm{supp}(\mu ), XY = YX} \mu (X) \mu (Y) < \varepsilon \end{aligned}\) or there exists some finite set \(\mathcal {S}\) contained in a 2-dimensional subspace of \(\text {Mat}_2(\mathbb {F})\) such that \(\mu (\mathcal {S}) \ge \varepsilon /8\) . This is sharp up to the multiplicative constant. We prove quantitatively stronger results when \(\mathbb {F} = \mathbb {C}\) and \( \mu \bigg ( \begin{pmatrix} a_1 & a_2\\ a_3 & a_4 \end{pmatrix} \bigg ) = \nu (a_1) \dots \nu (a_4) \ \ \text {for every} \ a_1, \dots , a_4 \in \mathbb {C}, \) with \(\nu \) being some finitely–supported probability measure on \(\mathbb {C}\) . For instance, when \(\mathcal {A} \subset \mathbb {R}\) is a generalised arithmetic progression or multiplicative progression of dimension d and \(\nu = \mathbbm {1}_{\mathcal {A}}/|\mathcal {A}|\) , our techniques imply that \(|\mathcal {A}|^{-3} \ll _d T(\mu ) \ll _d |\mathcal {A}|^{-3}\) . 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 \(\mathbb {C}\) . The latter includes applications of Schmidt’s subspace theorem and the resolution of the weak polynomial Freiman–Ruzsa conjecture over integers.
