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

Abstract

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

Light Corrections. The parameter h in the definition of T had to be a lot larger