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We obtain a series of estimates on the number of small integers and smallorder 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 thesimultaneous 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 thediscrete logarithm.

Distribution on elements of cosets of small subgroups and applications