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On Falconer's distance set problem in the plane

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arxiv 1808.09346 v1 pith:ONCGLC56 submitted 2018-08-28 math.CA

classification math.CA
keywords compactdimensiondistancedistancesfalconergreaterhausdorfflebesgue
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abstract

If $E \subset \mathbb{R}^2$ is a compact set of Hausdorff dimension greater than $5/4$, we prove that there is a point $x \in E$ so that the set of distances $\{ |x-y| \}_{y \in E}$ has positive Lebesgue measure.

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Cited by 2 Pith papers

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

  1. Upper bounds for Fourier decay rates of fractal measures

    math.CA 2019-08 accept novelty 7.0 of 10

    New counterexample constructions yield the exact parabolic Fourier decay rate (d−1)α/d for α in [d−1,d).

  2. Bisector energy and pinned distances in positive characteristic

    math.CO 2019-08 conditional novelty 6.0 of 10

    For A in F_q^2 with |A| at most p^{4/3}, some point of A determines Ω(|A|^{2/3}) distinct distances, improving the earlier Ω(|A|^{20/37}) bound.

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