Small solutions of diagonal congruences

Abstract

We prove that for $k \geq 2$, $0 <\varepsilon< \frac 1{k(k-1)}$, $n>\frac {k-1}{\varepsilon }$, prime $p> P(\varepsilon, k)$, and integers $c,a_i$, with $p \nmid a_i$, $1 \le i \le n$, there exists a solution $\underline{x}$ to the congruence $$ \sum_{i=1}^n a_ix_i^k \equiv c \mod p $$ in any cube $\mathcal{B}$ of side length $b \ge p^{\frac 1k + \varepsilon}$. Various refinements are given for smaller $n$ and for cubes centered at the origin.