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

Ernie Croot
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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.

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Ernie Croot. 2004-10-21. Sums of the Form 1/x_1^k + ... + 1/x_n^k Modulo a Prime. https://arxiv.org/abs/math/0403360

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