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
SN - 1073-7928
VL - 2008
SP - 1
EP - 29
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 1
M1 - rnn090
ER -