Product Sets of Rationals, Multiplicative Translates of Subgroups in Residue Rings, and Fixed Points of the Discrete Logarithm

被引：33

作者：

Bourgain, Jean ^{[2
]}

Konyagin, Sergei V. ^{[3
]}

Shparlinski, Igor E. ^{[1
]}

机构：

[1] Macquarie Univ, Dept Comp, Sydney, NSW 2109, Australia

[2] Inst Adv Study, Princeton, NJ 08540 USA

[3] Moscow MV Lomonosov State Univ, Dept Mech & Math, Moscow 119992, Russia

来源：

INTERNATIONAL MATHEMATICS RESEARCH NOTICES | 2008年 / 2008卷

基金：

澳大利亚研究理事会;

关键词：

D O I：

10.1093/imrn/rnn090

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We give a lower bound on the size of the product set of two arbitrary subsets of the set of Farey fractions of a given order and apply it to study the distribution of elements of multiplicative groups in residue rings. For example, we prove a conjecture of J. Holden and P. Moree on the behavior of the number of solutions to the congruence g(h) = h(mod p), 1 <= g, h <= p-1, on average over primes p. This congruence appears in studying fixed points of the discrete logarithm.

引用

页数：29

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