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New improvement to Falconer distance set problem in higher dimensions

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arxiv 2309.04103 v2 pith:AOC4FQE5 submitted 2023-09-08 math.CA math.COmath.MG

Xiumin Du , Yumeng Ou , Kevin Ren , Ruixiang Zhang This is my paper · ORCID
classification math.CAmath.COmath.MG
keywords fracboundsdimensionsdistancedimensionhausdorffimprovespinned
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abstract

We show that if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then there is a point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has positive Lebesgue measure. This improves upon bounds of Du-Zhang and Du-Iosevich-Ou-Wang-Zhang in all dimensions $d \ge 3$. We also prove lower bounds for Hausdorff dimension of pinned distance sets when $\dim_H (E) \in (\frac{d}{2} - \frac{1}{4} - \frac{3}{8d+4}, \frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4})$, which improves upon bounds of Harris and Wang-Zheng in dimensions $d \ge 3$.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates

    math.CA 2026-03 unverdicted novelty 8.0 of 10

    Under dim_H E >1, dim_H E + dim_H F >2 and F regular (equal Hausdorff and packing dimensions), there exists y in F such that the pinned distance set Δ_y(E) has positive Lebesgue measure.

  2. Pinned nonempty interior and volumes of simplices

    math.CA 2026-07 conditional novelty 7.0 of 10

    Doubly pinned nonempty interior for k-volume sets holds when dim_H(E)>(d+k-1)/2, via a cylinder-averaging triangle-area estimate and projection reduction.

  3. From weighted paraboloid restriction to $k$-stars and distance graphs

    math.CA 2026-07 accept novelty 7.0 of 10

    Pinned k-star distance sets of E have positive k-measure once dim(E) exceeds (n^{2}+nk+k)/(2n+1), via a weighted paraboloid Fourier-extension identity.

  4. Random Constructions for Sharp Estimates of Mizohata-Takeuchi Type

    math.CA 2025-06 conditional novelty 6.0 of 10

    A random construction of weights yields, with high probability, sharp epsilon-loss Mizohata-Takeuchi estimates for the Fourier extension operator.

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