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1 April 2001 Linear equations over $\mathbb{F}_p$ and moments of exponential sums
Vsevolod F. Lev
Duke Math. J. 107(2): 239-263 (1 April 2001). DOI: 10.1215/S0012-7094-01-10722-9

## Abstract

Two of our principal results (in a simplified form) are as follows.

THEOREM

For $p$ prime, the number of solutions of the equation $$c_{1}a_{1}+\cdots+c_{k}a_ {k}=\lambda,\quad a_{j}\in A_{j},$$ where $c_j ∈ \mathbb{F}_p^x$ and $λ ∈\mathbb{F}_p$ are fixed coefficients, and the variables $a_j$ range over sets $A_j ⊆ \mathbb{F}_p$, does not exceed the number of solutions of the equation $$a_{1}+\cdots+a^{k}=0,\quad a_ {j}\in\overline {A}_{j},$$ where the variables $a_j$ range over arithmetic progressions $\overline {A}_j⊆\mathbb{F}_p$ of cardinalities $|\overline {A}_j|=|A_j|$, balanced around zero.

This readily implies an integer version, which strengthens a result of R. Gabriel, G. Hardy, and J. Littlewood.

THEOREM

Let $A$ be a set of $n=|A|$ residues modulo a prime $p$. For $z∈\mathbb{F}_p$, write $$S_{A}(z)=\sum_{a\in A}e^{2\pi i(az/p)}.$$ Then for $ε>0$ we have $$\#\lbrace z\in\mathbb {F}_{p}^{x} : |S_{A}(z)>(1-\varepsilon)n\rbrace\leq\frac {2\sqrt {6}}{\pi}\frac {p}{n}\varepsilon^{1/2}(1+o(1)),$$ provided $n→∞$ and $ε→0$. Equality is attained when $A$ is an arithmetic progression modulo $p$.

This implies an integer version that is due to A. A. Yudin.

## Citation

Vsevolod F. Lev. "Linear equations over $\mathbb{F}_p$ and moments of exponential sums." Duke Math. J. 107 (2) 239 - 263, 1 April 2001. https://doi.org/10.1215/S0012-7094-01-10722-9

## Information

Published: 1 April 2001
First available in Project Euclid: 5 August 2004

zbMATH: 1035.11007
MathSciNet: MR1823048
Digital Object Identifier: 10.1215/S0012-7094-01-10722-9

Subjects:
Primary: 11B75
Secondary: 11D04 , 11L07 , 42C20  