Combinational properties of sets of residues modulo a prime and the Erdős-Graham problem (Q881006)

It is proved that for every \(\varepsilon >0\) and sufficiently large prime \(p\) for every residue class \(a\) modulo \(p\) there are integers \(0<x_1 < \dots <x_N <p^\varepsilon \), \(N = O(\varepsilon ^{-2})\) such that \(\sum x_i^{-1} \equiv a \pmod{p} \). This improves a result of Shparlinski where the bound on the number of summands was \(O(\varepsilon ^{-3})\).