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- W3125537384 abstract "Let $mathbb{F}_q$ be an arbitrary finite field, and $mathcal{E}$ be a set of points in $mathbb{F}_q^d$. Let $Delta(mathcal{E})$ be the set of distances determined by pairs of points in $mathcal{E}$. By using the Kloosterman sums, Iosevich and Rudnev proved that if $|mathcal{E}|ge 4q^{frac{d+1}{2}}$, then $Delta(mathcal{E})=mathbb{F}_q$. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. In this paper, we use the recent point-plane incidence bound due to Rudnev to prove that if $mathcal{E}$ has Cartesian product structure in vector spaces over prime fields, then we can break the exponent $(d+1)/2$, and still cover all distances. We also show that the number of pairs of points in $mathcal{E}$ of any given distance is close to its expected value." @default.
- W3125537384 created "2021-02-01" @default.
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- W3125537384 date "2020-12-31" @default.
- W3125537384 modified "2023-09-27" @default.
- W3125537384 title "Distribution of distances in positive characteristic" @default.
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- W3125537384 doi "https://doi.org/10.2140/pjm.2020.309.437" @default.
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