On some congruences and exponential sums

Abstract

Let $\epsilon >0$ be a fixed small constant, ${\mathbb{F}}_p$ be the finite field of $p$ elements for prime $p$. We consider additive and multiplicative problems in ${\mathbb{F}}_p$ that involve intervals and arbitrary sets. Representative examples of our results are as follows. Let $\mathcal{M}$ be an arbitrary subset of ${\mathbb{F}}_p$. If $\#\mathcal{M}>p^{1/3+\epsilon }$ and $H\ge p^{2/3}$ or if $\#\mathcal{M}>p^{3/5+\epsilon }$ and $H\ge p^{3/5+\epsilon }$ then all, but $O(p^{1-\delta })$ elements of ${\mathbb{F}}_p$ can be represented in the form $hm$ with $h\in [1,H]$ and $m\in \mathcal{M}$, where $\delta >0$ depends only on $\epsilon$. Furthermore, let $\mathcal{X}$ be an arbitrary interval of length $H$ and $s$ be a fixed positive integer. If $$H>p^{17/35+\epsilon },\#\mathcal{M}>p^{17/35+\epsilon }.$$ then the number $T_6(\lambda )$ of solutions of the congruence $$\frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}+\frac{m_4}{x_4^s}+\frac{m_5}{x_5^s}+\frac{m_6}{x_6^s}\equiv \lambda \mathrm{mod}p,m_i\in \mathcal{M},x_i\in \mathcal{X},i=1,\dots ,6,$$ satisfies $$T_6(\lambda )=\frac{H^6(\#\mathcal{M}{)}^6}{p}\left(1+O(p^{-\delta })\right),$$ where $\delta >0$ depends only on $s$ and $\epsilon$.