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An improved result for Falconer’s distance set problem in even dimensions

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Abstract

We show that if compact set \(E\subset \mathbb {R}^d\) has Hausdorff dimension larger than \(\frac{d}{2}+\frac{1}{4}\), where \(d\ge 4\) is an even integer, then the distance set of E has positive Lebesgue measure. This improves the previously best known result towards Falconer’s distance set conjecture in even dimensions.

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Acknowledgements

XD is supported by NSF DMS-1856475. AI was partially supported by NSF HDR TRIPODS 1934985. YO is supported by NSF DMS-1854148. HW is funded by the S.S. Chern Foundation and NSF DMS-1638352. RZ is supported by NSF DMS-1856541. We would like to thank Pablo Shmerkin for pointing out a minor issue in a previous version regarding the pushforward measure under the orthogonal projection.

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Authors and Affiliations

  1. Department of Mathematics, Northwestern University, Evanston, IL, 60208, USA

    Xiumin Du

  2. Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

    Alex Iosevich

  3. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Yumeng Ou

  4. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

    Hong Wang & Ruixiang Zhang

  5. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Ruixiang Zhang

Authors
  1. Xiumin Du
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  2. Alex Iosevich
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  3. Yumeng Ou
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  4. Hong Wang
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  5. Ruixiang Zhang
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Correspondence to Ruixiang Zhang.

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Communicated by Loukas Grafakos.

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Du, X., Iosevich, A., Ou, Y. et al. An improved result for Falconer’s distance set problem in even dimensions. Math. Ann. 380, 1215–1231 (2021). https://doi.org/10.1007/s00208-021-02170-1

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  • DOI: https://doi.org/10.1007/s00208-021-02170-1

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