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# The Large Sieve Inequality for the Exponential Sequence $\lambda^{[O(n^{15/14+o(1)})]}$ Modulo Primes

Let $\lambda$ be a fixed integer exceeding $1$ and $s_n$ 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 obtain nontrivial estimates of the associated trigonometric sums on average" and establish equidistribution properties of the numbers $\lambda^{s_n} , n\le p(\log p)^{2+\varepsilon}$, modulo $p$ for most primes $p.$