On Mattila's intersection problem in the plane: sharp dimension and positive measure of rotated difference sets
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Pham, Thang, Shen, Chun-Yen, Tran, Dung The
Institutions: Xidian University, National Taiwan University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221833282