The distribution of solutions to equations over finite fields

Abstract:

Let ${\mathbb {F}_q}$ be the finite field in $q = {p^f}$ elements, $\underline F (\underline x )$ be a $k$-tuple of polynomials in ${\mathbb {F}_q}[{x_1}, \ldots ,{x_n}]$, $V$ be the set of points in $\mathbb {F}_q^n$ satisfying $\underline F (\underline x ) = \underline 0$ and $S$, $T$ be any subsets of $\mathbb {F}_q^n$. Set $\phi (V,\underline 0 ) = |V| - {q^{n - k}}$, \[ \phi (V,\underline y ) = \sum \limits _{\underline x \in V} {e\left ( {\frac {{2\pi i}} {p}\operatorname {Tr} (\underline x \cdot \underline y )} \right )\quad {\text {for}}\;\underline y \ne \underline 0 ,} \] and $\Phi (V) = {\max _{\underline y }}|\phi (V,\underline y )|$. We use finite Fourier series to show that $(S + T) \cap V$ is nonempty if $|S||T| > {\Phi ^2}(V){q^{2k}}$. In case $q = p$ we deduce from this, for example, that if $C$ is a convex subset of ${\mathbb {R}^n}$ symmetric about a point in ${\mathbb {Z}^n}$, of diameter $< 2p$ (with respect to the sup norm), and $\operatorname {Vol} (C) > {2^{2n}}\Phi (V){p^k}$, then $C$ contains a solution of $\underline F (\underline x ) \equiv \underline 0 (\bmod p)$. We also show that if $B$ is a box of points in $\mathbb {F}_q^n$ not contained in any $(n - 1)$-dimensional subspace and $|B| > 4 \cdot {2^{nf}}\Phi (V){q^k}$, then $B \cap V$ contains $n$ linearly independent points.