#### Vol. 309, No. 2, 2020

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Distribution of distances in positive characteristic

### Thang Pham and Lê Anh Vinh

Vol. 309 (2020), No. 2, 437–451
##### Abstract

Let ${\mathbb{𝔽}}_{q}$ be an arbitrary finite field, and $\mathsc{ℰ}$ be a point set in ${\mathbb{𝔽}}_{q}^{d}$. Let $\Delta \left(\mathsc{ℰ}\right)$ be the set of distances determined by pairs of points in $\mathsc{ℰ}$. Using Kloosterman sums, Iosevich and Rudnev (2007) proved that if $|\mathsc{ℰ}|\ge 4{q}^{\left(d+1\right)∕2}$ then $\Delta \left(\mathsc{ℰ}\right)={\mathbb{𝔽}}_{q}$. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. We use the point-plane incidence bound due to Rudnev to prove that if $\mathsc{ℰ}$ has Cartesian product structure in vector spaces over prime fields, then we can break the exponent $\left(d+1\right)∕2$ and still cover all distances. We also show that the number of pairs of points in $\mathsc{ℰ}$ of any given distance is close to its expected value.

##### Keywords
distances, finite fields, incidence, Rudnev's point-plane incidence bound
##### Mathematical Subject Classification
Primary: 14N10, 51A45, 52C10