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

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