On the density or measure of sets and their sumsets in the integers or the circle

Let d(A) be the asymptotic density (if it exists) of a sequence of integers A. For any real numbers 0 <= alpha <= beta <= 1, we solve the question of the existence of a sequence A of positive integers such that d(A) =alpha and d(A + A) = beta. More generally we study the set of k-tuples (d(iA))(1 <= i <= k) for A subset of N. This leads us to introduce subsets defined by diophantine constraints inside a random set of integers known as the set of "pseudo sth powers". We consider similar problems for subsets of the circle R/Z, that is, we partially determine the set of k-tuples (mu(iA))(1 <= i <= k )for A subset of R/Z. ( )(C) 2019 Elsevier Inc. All rights reserved.