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

Abstract

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 $${\lambda }^{s_n},n\le X(\log X{)}^{2+\epsilon }$$ are uniformly distributed modulo $p.$

19 pages; in the new version we clean up some estimates and add several more consequences of the main result