Positive exponential sums, difference sets and recurrence

We use the positive exponential sums method (also known as Delsarteʼs method) to find explicit upper bounds related to difference set property of sets of integers. Specifically, we find asymptotic upper bounds on the cardinality of a set of integers A so that A − A does not intersect a given set D. We apply it to sets D of perfect squares, sets of values of odd polynomials, and shifted primes p − 1 . We also develop an ergodic-theoretical interpretation of the bounds.