TY - JOUR

T1 - Product sets of rationals, multiplicative translates of subgroups in residue rings, and fixed points of the discrete logarithm

AU - Bourgain, Jean

AU - Konyagin, Sergei V.

AU - Shparlinski, Igor E.

N1 - Corrigendum can be found in International Mathematics Research Notices, Volume 2009(16), 3146-3147, http://dx.doi.org/10.1093/imrn/rnp041

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77952241082&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/record.url?scp=84864146205&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnn090

DO - 10.1093/imrn/rnn090

M3 - Article

AN - SCOPUS:77952241082

VL - 2008

SP - 1

EP - 29

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 1

M1 - rnn090

ER -