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PreprintCC BY 4.0arXiv

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⊂R2 be nonempty Borel sets. We prove that dimH(gA−B)≥min{dimH A+dimH B,2} for almost all g∈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 LL2(gA−B)>0 for almost all g if both dimensions exceed (√(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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Cite

BibTeXRIS
BibTeX
@misc{pham2026mattilas,
  title = {{On Mattila's intersection problem in the plane: sharp dimension and positive measure of rotated difference sets}},
  author = {Thang Pham and Chun-Yen Shen and Dung The Tran},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16114},
  eprint = {2610.16114},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16114},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Mattila's intersection problem in the plane: sharp dimension and positive measure of rotated difference sets
AU  - Thang Pham
AU  - Chun-Yen Shen
AU  - Dung The Tran
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16114
UR  - https://arxiv.org/abs/2610.16114
AB  - Let A,B⊂R^2 be nonempty Borel sets. We prove that dimH(gA−B)≥min{dimH A+dimH B,2} for almost all g∈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 (√(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.
ER  -