Sums of the Form 1/x_1^k + ... + 1/x_n^k Modulo a Prime

Abstract

We show that for every $0 < ε\leq 1$ and integer $k\geq 1$, there exists an integer $n = n(ε,k)$ so that for all primes $p$, and integers $0 \leq a \leq p-1$, there exist integers $1 \leq x_1 < ... < x_n \leq p^ε$ such that $a \equiv x_1^{-1} + ... + x_n^{-1} \pmod{p}$. This extends a result of I. Shparlinski.