# On the exponential large sieve inequality for sparse sequences modulo primes

## TL;DR

This paper improves the large sieve inequality for sparse exponential sequences modulo primes, extending its applicability to more rapidly increasing sequences and applying it to digit-prescribed integers.

## Contribution

It introduces a stronger version of the large sieve inequality for sparse exponential sequences, surpassing previous bounds and enabling new applications.

## Key findings

- Enhanced inequality valid for sequences with s_n ≤ n^{2+o(1)}
- Applicable to sequences with faster growth than previous bounds
- Applied to arithmetic properties of integers with prescribed digits

## Abstract

We complement the argument of M. Z. Garaev (2009) with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form $\lambda^{s_n}$. In particular, we obtain a result which is non-trivial for monotonically increasing sequences $\cal{S}=\{s_n \}_{n=1}^{\infty}$ provided $s_n\le n^{2+o(1)}$, whereas the original argument of M. Z. Garaev requires $s_n \le n^{15/14 +o(1)}$ in the same setting. We also give an application of our result to arithmetic properties of integers with almost all digits prescribed.