Abstract: In this talk I will present a classical problem of Vinogradov about the distribution of primes in the subsets of integers satisfying certain restrictions on the fractional part of n^a for non-integer a > 0 and in the intersection of this sets with arithmetic progressions. We show how this problem is related to exponential sums over primes and describe some of the approaches to estimating such sums.
The recent progress towards these questions was made in the following two works: