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