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 $\varepsilon$.
Categories: math.NT