Let $p$ be a prime. Let $F_p$ be the finite field of $p$ elements. Let $A$ be a subset of $F_p$ of size $s$. Assume that $s > 2$ is polylogarithmic in $p$.

Suppose that we want to count number of solutions of $$ x_1 + x_2 + ... + x_k = t (\mod\ p) $$ under the restriction that $x_i \in A $ for all $1\leq i\leq k$. The expectation seems to be $s^k/p$ if $k$ is large, independent of $t$.

My question is: Does there exist $A$ such that the expectation is known to be quite sharp (namely that the error term is poly(k p) in absolute value)?

Thanks a lot