The distribution of solutions to equations over finite fields

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 $\phi (V,\underline 0 ) = \vert V\vert - {q^{n - k}}$,

$$\phi (V,\underline y ) = \sum\limits_{\underline x \in V} {e\left... ...ot \underline y )} \right)\quad {\text{for}}\;\underline y \ne \underline 0 ,}$$

and $\Phi (V) = {\max _{\underline y }}\vert\phi (V,\underline y )\vert$. We use finite Fourier series to show that $(S + T) \cap V$ is nonempty if $\vert S\vert\vert T\vert > {\Phi ^2}(V){q^{2k}}$. 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 $\vert B\vert > 4 \cdot {2^{nf}}\Phi (V){q^k}$, then $B \cap V$ contains $n$ linearly independent points.