On the exponential large sieve inequality for sparse sequences modulo primes

We complement the argument of M. Z. Garaev (2009) [9] with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form λsn. In particular, we obtain a result which is non-trivial for monotonically increasing sequences S={sn}n=1∞ provided sn⩽n2+o(1), whereas the original argument of M. Z. Garaev requires sn⩽n15/14+o(1) in the same setting. We also give an application of our result to arithmetic properties of integers with almost all digits prescribed.