Geodesic
Combinational properties of sets of residues modulo a prime and the Erd\H os--Graham problem
Matematičeskie zametki, Tome 79 (2006) no. 3, pp. 384-395

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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. 
@article{MZM_2006_79_3_a5,
     author = {A. A. Glibichuk},
     title = {Combinational properties of sets of residues modulo a prime and the {Erd\H} {os--Graham} problem},
     journal = {Matemati\v{c}eskie zametki},
     pages = {384--395},
     publisher = {mathdoc},
     volume = {79},
     number = {3},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_79_3_a5/}
}
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A. A. Glibichuk. Combinational properties of sets of residues modulo a prime and the Erd\H os--Graham problem. Matematičeskie zametki, Tome 79 (2006) no. 3, pp. 384-395. http://geodesic.mathdoc.fr/item/MZM_2006_79_3_a5/