Geodesic
Combinational properties of sets of residues modulo a prime and the Erdős–Graham problem
Matematičeskie zametki, Tome 79 (2006) no. 3, pp. 384-395 
A. A. Glibichuk. Combinational properties of sets of residues modulo a prime and the Erdős–Graham problem. Matematičeskie zametki, Tome 79 (2006) no. 3, pp. 384-395. http://geodesic.mathdoc.fr/item/MZM_2006_79_3_a5/
@article{MZM_2006_79_3_a5,
     author = {A. A. Glibichuk},
     title = {Combinational properties of sets of residues modulo a prime and the {Erd\H{o}s{\textendash}Graham} problem},
     journal = {Matemati\v{c}eskie zametki},
     pages = {384--395},
     year = {2006},
     volume = {79},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_79_3_a5/}
}
TY  - JOUR
AU  - A. A. Glibichuk
TI  - Combinational properties of sets of residues modulo a prime and the Erdős–Graham problem
JO  - Matematičeskie zametki
PY  - 2006
SP  - 384
EP  - 395
VL  - 79
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MZM_2006_79_3_a5/
LA  - ru
ID  - MZM_2006_79_3_a5
ER  - 
%0 Journal Article
%A A. A. Glibichuk
%T Combinational properties of sets of residues modulo a prime and the Erdős–Graham problem
%J Matematičeskie zametki
%D 2006
%P 384-395
%V 79
%N 3
%U http://geodesic.mathdoc.fr/item/MZM_2006_79_3_a5/
%G ru
%F MZM_2006_79_3_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Erdős P., Graham R. L., Old and New Problems and Results In Combinational Number Theory, Monograph. Enseign. Math., 28, 1980

[2] Croot E. S., “On some questions of Erdős and Graham about Egyptian fractions”, Mathematika, 46 (1999), 359–372 | MR | Zbl

[3] Sparlinski I. E., “On a question of Erdős and Graham”, Arch. Math., 78 (2002), 445–448 | DOI | MR

[4] Karatsuba A. A., “Raspredelenie obratnykh velichin v koltse vychetov po zadannomu modulyu”, Dokl. RAN, 333:2 (1993), 138–139 | Zbl

[5] Karatsuba A. A., “Drobnye doli spetsialnogo vida funktsii”, Izv. RAN. Seriya matem., 59:4 (1995), 61–80 | MR | Zbl

[6] Karatsuba A. A., “Analogi nepolnykh summ Kloostermana i ikh prilozheniya”, Tatra Mountain Math. Publ., 11 (1997), 89–120 | MR | Zbl

[7] Croot E. S., Reciprocal power sums modulio a prime, , 2004 E-print math.NT/0403360

[8] Bourgain J., Katz N., Tao T., A sum-product estimate in finite fields and their applications, , 2003 E-print math.CO/0301343

[9] Shkredov I. D., “O nekotorykh additivnykh zadachakh, svyazannykh s pokazatelnoi funktsiei”, UMN, 58:4 (2003), 165–166 | MR | Zbl

[10] Karatsuba A. A., “Additivnye sravneniya”, Izv. RAN. Seriya matem., 61:2 (1997), 81–94 | MR | Zbl

[11] Karatsuba A. A., “O pravilnykh mnozhestvakh v ostatochnykh klassakh”, Matem. zametki, 64:2 (1998), 224–228 | MR | Zbl