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Distribution on elements of cosets of small subgroups and applications

By Jean Bourgain, Sergei Konyagin and Igor Shparlinski


We obtain a series of estimates on the number of small integers and small order Farey fractions which belong to a given coset of a subgroup of order $t$ of the group of units of the residue ring modulo a prime $p$, in the case when $t$ is small compared to $p$. We give two applications of these results: to the simultaneous distribution of two high degree monomials $x^{k_1}$ and $x^{k_2}$ modulo $p$ and to a question of J.Holden and P.Moree on fixed points of the discrete logarithm.Comment: 35 page

Topics: Mathematics - Number Theory, 11A15, 11L07, 11N25
Year: 2011
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