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arXiv · 2610.16114

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

Thang Pham·Chun-Yen Shen·Dung The Tran
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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.

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Thang Pham, Chun-Yen Shen, Dung The Tran. 2026-10-12. On Mattila's intersection problem in the plane: sharp dimension and positive measure of rotated difference sets. https://arxiv.org/abs/2610.16114

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