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


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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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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Correspondence to Ruixiang Zhang.

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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. (2021).

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