Abstract

Let ${\mathbb{F}}_q$ be a finite field of order $q$. We prove that if $d\ge 2$ is even and $E\subset {\mathbb{F}}_q^d$ with $\left|E\right|\ge 9q^{d∕2}$ then

where

If the dimension $d$ is odd and $E\subset {\mathbb{F}}_q^d$ with $\left|E\right|\ge 6q^{d∕2}$, then

where ${\mathbb{F}}_q^{+}$ denotes the set of nonzero quadratic residues in ${\mathbb{F}}_q$. Both results are, in general, best possible, including the conclusion about the nonzero quadratic residues in odd dimensions.

Keywords

Mathematical Subject Classification 2010

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