On congruences with products of variables from short intervals and applications

Jean Bourgain; Moubariz Z. Garaev; Sergei V. Konyagin; Igor E. Shparlinski

doi:10.1134/S0081543813010057

On congruences with products of variables from short intervals and applications

Jean Bourgain^*, Moubariz Z. Garaev, Sergei V. Konyagin, Igor E. Shparlinski

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

30 Citations (Scopus)

Abstract

We obtain upper bounds on the number of solutions to congruences of the type (x₁ + s)... (x_ν + s) ≡ (y₁ + s)... (x_ν + s) ≢ 0 (mod p) modulo a prime p with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function.

Original language	English
Pages (from-to)	61-90
Number of pages	30
Journal	Proceedings of the Steklov Institute of Mathematics
Volume	280
Issue number	1
DOIs	https://doi.org/10.1134/S0081543813010057
Publication status	Published - 2013

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AU - Garaev, Moubariz Z.

AU - Konyagin, Sergei V.

AU - Shparlinski, Igor E.

PY - 2013

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AB - We obtain upper bounds on the number of solutions to congruences of the type (x1 + s)... (xν + s) ≡ (y1 + s)... (xν + s) ≢ 0 (mod p) modulo a prime p with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function.

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On congruences with products of variables from short intervals and applications

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