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On Falconer's distance set problem in the plane
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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.
Forward citations
Cited by 2 Pith papers
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Upper bounds for Fourier decay rates of fractal measures
New counterexample constructions yield the exact parabolic Fourier decay rate (d−1)α/d for α in [d−1,d).
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Bisector energy and pinned distances in positive characteristic
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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