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arXiv · 2403.05078

Distribution of solutions to systems of congruences in balls

Michael Harm
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Abstract

Let $G_1,\dots, G_n\in \mathbb{F}_p[X_1,\dots,X_m]$ be $n$ polynomials in $m$ variables over the finite field $\mathbb{F}_p$ of $p$ elements. For any sufficiently large prime $p$ and non-trivial bounds for the Weyl sums associated to the non-trivial linear combinations of $G=(G_1,\dots, G_n)$, we study various properties regarding the distribution of the vectors by fractional parts \begin{equation*} \bigg(\bigg\{ \frac{G_1(\textbf{x})}{p}\bigg\},\cdots,\bigg\{ \frac{G_n(\textbf{x})}{p}\bigg\}\bigg)\in \mathbb{T}^n,\hspace{10pt} \textbf{x}\in \mathbb{F}_p^m. \end{equation*} We prove refinements of equidistribution, such as bounds for the ball discrepancy and variance.

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Michael Harm. 2024-03-08. Distribution of solutions to systems of congruences in balls. https://doi.org/10.1016/j.ffa.2026.102834

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