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Weighted refined decoupling estimates and application to Falconer distance set problem

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arxiv 2309.04501 v1 pith:4TCKYHXS 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 applicationdecouplingdistanceestimatesfracrefinedweightedfalconer
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abstract

We prove some weighted refined decoupling estimates. As an application, we give an alternative proof of the following result on Falconer's distance set problem by the authors in a companion work: 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 $d\geq 4$, then there is a point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has positive Lebesgue measure. Aside from this application, the weighted refined decoupling estimates may be of independent interest.

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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. Mizohata-Takeuchi inequalities for orthonormal systems

    math.CA 2025-06 conditional novelty 8.0 of 10

    Orthonormal systems of inputs satisfy a global Mizohata-Takeuchi-type weighted inequality for the sphere and paraboloid, with an X-ray transform norm over the midpoint set K^diamond on the right-hand side.

  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. Additive structures imply more distances in $\mathbb{F}_q^d$

    math.CO 2025-10 conditional novelty 6.0 of 10

    For (4,s)-Salem sets in F_q^d, the threshold for determining a positive proportion of all distances is improved to q^{min{(d+2)/(4s+1),(d+4)/(8s)}}.

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