On some congruences and exponential sums

Abstract

Let $\varepsilon>0$ 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 $\#{\mathcal M} >p^{1/3+\varepsilon}$ and $H\ge p^{2/3}$ or if $\#{\mathcal M} >p^{3/5+\varepsilon}$ and $H\ge p^{3/5+\varepsilon}$ 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 $\varepsilon$. Furthermore, let $ {\mathcal X}$ be an arbitrary interval of length $H$ and $s$ be a fixed positive integer. If $$ H> p^{17/35+\varepsilon}, \quad \#{\mathcal M} > p^{17/35+\varepsilon}. $$ then the number $T_6(\lambda)$ of solutions of the congruence $$ \frac{m_1}{x_1^s}+ \frac{m_2}{x_2^s}+ \frac{m_3}{x_3^s}+\frac{m_4}{x_4^s}+ \frac{m_5}{x_5^s}+\frac{m_6}{x_6^s} \equiv \lambda\mod p, \qquad m_i\in {\mathcal M}, \quad \ x_i \in {\mathcal X}, \quad i =1, \ldots, 6, $$ 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 $\varepsilon$.