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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2403.05078 |
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| _version_ | 1866916023421108224 |
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| author | Harm, Michael |
| author_facet | Harm, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05078 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of solutions to systems of congruences in balls Harm, Michael Number Theory 11D79, 11K38, 11T23 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. |
| title | Distribution of solutions to systems of congruences in balls |
| topic | Number Theory 11D79, 11K38, 11T23 |
| url | https://arxiv.org/abs/2403.05078 |