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

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