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- W4380136761 abstract "In this paper we study the following variant of the Falconer distance problem. Let $E$ be a compact subset of ${mathbb{R}}^d$, $d ge 1$, and define $$ Box(E)=left{sqrt{{|x-y|}^2+{|x-z|}^2}: x,y,z in E,, yneq z right}.$$ We shall prove using a variety of methods that if the Hausdorff dimension of $E$ is greater than $frac{d}{2}+frac{1}{4}$, then the Lebesgue measure of $Box(E)$ is positive. This problem can be viewed as a singular variant of the classical Falconer distance problem because considering the diagonal $(x,x)$ in the definition of $Box(E)$ poses interesting complications stemming from the fact that the set ${(x,x): x in E}subseteq mathbb{R}^{2d}$ is much smaller than the sets for which the Falconer type results are typically established. We also prove a finite field variant of the Euclidean results for $Box(E)$ and indicate both the similarities and the differences between the two settings." @default.
- W4380136761 created "2023-06-10" @default.
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- W4380136761 date "2023-06-08" @default.
- W4380136761 modified "2023-10-12" @default.
- W4380136761 title "A singular variant of the Falconer distance problem" @default.
- W4380136761 doi "https://doi.org/10.48550/arxiv.2306.05247" @default.
- W4380136761 hasPublicationYear "2023" @default.
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