Combinational properties of sets of residues modulo a prime and the Erd\H os--Graham problem

A. A. Glibichuk

Matematičeskie zametki, Tome 79 (2006) no. 3, pp. 384-395

Résumé

Consider an arbitrary $\varepsilon>0$ and a sufficiently large prime $p>2$. It is proved that, for any integer $a$, there exist pairwise distinct integers $x_1,x_2,\dots,x_N$, where $N=8([1/\varepsilon+1/2]+1)^2$ such that $1\le x_i\le p^\varepsilon$, $i=1,\dots,N$, and $$ a\equiv x_1^{-1}+\dotsb+x_N^{-1}\pmod p, $$ where $x_i^{-1}$ is the least positive integer satisfying $x_i^{-1}x_i\equiv1\pmod p$. This improves a result of Sparlinski.