On Mattila's intersection problem in the plane: sharp dimension and positive measure of rotated difference sets

Let $A,B\subset {\mathbb{R}}^2$ be nonempty Borel sets. We prove that $$\text{dimH}(gA-B)\ge \min \{\text{dimH}A+\text{dimH}B,2\}$$ for almost all $g\in \mathrm{O}(2)$. This bound is sharp in general, and the proof gives a corresponding statement for the lower Hausdorff dimension of convolutions of Frostman measures. We also prove that ${\text{LL}}^2(gA-B)>0$ for almost all $g$ if both dimensions exceed $(\sqrt{13}-1)\mathrm{/}2$. The second result follows from an angular decomposition, refined decoupling and a two-ends Furstenberg estimate. Some applications on intersection patterns are also addressed.