The large sieve inequality with exponential functions and the distribution of Mersenne numbers modulo primes

Let S be a set of T = # S positive integers in the interval [1,N] and let ϑ ≥ 2 be a fixed integer. We obtain a version of the large sieve inequality for the set ϑⁿ, n ∈ S. For example, if S is not too sparse, then the residues ϑⁿ(mod p) are uniformly distributed for almost all primes in the interval [Q/2, Q] for a very wide range of values of Q. In the special case, when S is the set of primes of the interval [1,N], we use a different approach, which in particular implies that Mersenne numbers are uniformly distributed modulo p for almost all p. This complements a series of recent bounds, which are considerably weaker but instead apply to “individual” values of p rather than “on average.”