Binary and ternary congruences involving intervals and sets modulo a prime

Abstract

Let $s$ be a fixed positive integer constant, $\epsilon$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{\mathcal{M}\subseteq {\mathbb{F}}_p^{\ast }$ of size $|\mathcal{M}|=\lfloor p^{14/29}\rfloor$ and integer $H=\lfloor p^{14/29+\epsilon }\rfloor$, any integer $\hat{I}»$ can be represented in the form $$\frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \hat{I}»\mathrm{mod}p,$$ with $$m_i\in \mathcal{M},1\le x_i\le H,i=1,2,3.$$ When $s=1$ we show that for almost all primes $p$ the following holds: if $|\mathcal{M}|=\lfloor p^{1/2}\rfloor$ and $H=\lfloor p^{1/2}(\log p{)}^{6+\epsilon }\rfloor$, then any integer $\hat{I}»$ can be represented in the form $$\frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \hat{I}»\mathrm{mod}p,$$ with $$m_i\in \mathcal{M},1\le x_i\le H,i=1,2.$$