Product sets of rationals, multiplicative translates of subgroups in residue rings, and fixed points of the discrete logarithm

Jean Bourgain*, Sergei V. Konyagin, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

33 Citations (Scopus)

Abstract

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 gh ≡ h (mod p), 1 ≤g, h ≤p-1, on average over primes p. This congruence appears in studying fixed points of the discrete logarithm.

Original languageEnglish
Article numberrnn090
Pages (from-to)1-29
Number of pages29
JournalInternational Mathematics Research Notices
Volume2008
Issue number1
DOIs
Publication statusPublished - 2008

Bibliographical note

Corrigendum can be found in International Mathematics Research Notices, Volume 2009(16), 3146-3147, http://dx.doi.org/10.1093/imrn/rnp041

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