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

Source abstract

Let $A,B\subset\R^2$ be nonempty Borel sets. We prove that \[ \dimH(gA-B)\ge\min\{\dimH A+\dimH B,2\} \] 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 \(\LL^2(gA-B)>0\) for almost all $g$ if both dimensions exceed $(\sqrt{13}-1)/2$. The second result follows from an angular decomposition, refined decoupling and a two-ends Furstenberg estimate. Some applications on intersection patterns are also addressed.