The large sieve for 2^{[O(n^(15/14+o(1)))]} modulo primes

Let $\lambda$ be a fixed integer, $\lambda\ge 2.$ Let $s_n$ be any strictly increasing sequence of positive integers satisfying $s_n\le n^{15/14+o(1)}.$ In this paper we give a version of the large sieve inequality for the sequence $\lambda^{s_n}.$ In particular, we prove that for $\pi(X)(1+o(1))$ primes $p, \ p\le X,$ the numbers $$ \lambda^{s_n},\quad n\le X(\log X)^{2+\epsilon} $$ are uniformly distributed modulo $p.$