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PreprintarXiv licencearXiv

Lebesgue measure of distance sets and L2 ball inflation

  • Shengwen Gan
  • Bochen Liu
  • Shukun Wu
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Abstract

We give an alternative proof of the 5/4 result of Guth-Iosevich-Ou-Wang on the Falconer distance conjecture: if a compact set E in the plane has Hausdorff dimension >5/4, then there exists x∈ E such that the pinned distance set Δx(E):={|x−y|: y∈ E} has positive Lebesgue measure. Moreover, we are able to beat the 5/4 barrier if the packing dimension of E is <2. More generally, we obtain sufficient conditions involving both the Hausdorff and packing dimensions of E. These conditions improve the 5/4 threshold when the packing dimension of E is <2, and recover the second author's result for dimH E=dimP E>1. The proof is based on L2 ball inflation and does not use decoupling inequalities.

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Cite

BibTeXRIS
BibTeX
@misc{gan2026lebesgue,
  title = {{Lebesgue measure of distance sets and $L^2$ ball inflation}},
  author = {Shengwen Gan and Bochen Liu and Shukun Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05015},
  eprint = {2610.05015},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05015},
}
RIS
TY  - UNPB
TI  - Lebesgue measure of distance sets and L^2 ball inflation
AU  - Shengwen Gan
AU  - Bochen Liu
AU  - Shukun Wu
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05015
UR  - https://arxiv.org/abs/2610.05015
AB  - We give an alternative proof of the 5/4 result of Guth-Iosevich-Ou-Wang on the Falconer distance conjecture: if a compact set E in the plane has Hausdorff dimension >5/4, then there exists x∈ E such that the pinned distance set Δ_x(E):={|x−y|: y∈ E} has positive Lebesgue measure. Moreover, we are able to beat the 5/4 barrier if the packing dimension of E is <2. More generally, we obtain sufficient conditions involving both the Hausdorff and packing dimensions of E. These conditions improve the 5/4 threshold when the packing dimension of E is <2, and recover the second author's result for dim_H E=dim_P E>1. The proof is based on L^2 ball inflation and does not use decoupling inequalities.
ER  -