Linear equations over $\mathbb{F}_p$ and moments of exponential sums

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.