  Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of
integers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve the
question of the existence of a sequence $A$ of positive integers such that
$\mathrm{d}(A)=\alpha$ and $\mathrm{d}(A+A)=\beta$. More generally we study the
set of $k$-tuples $(\mathrm{d}(iA))_{1\leq i\leq k}$ for $A\subset \mathbb{N}$.
This leads us to introduce subsets defined by diophantine constraints inside a
random set of integers known as the set of ``pseudo $s$th powers''. We consider
similar problems for subsets of the circle $\mathbb{R}/\mathbb{Z}$, that is, we
partially determine the set of $k$-tuples $(\mu(iA))_{1\leq i\leq k}$ for
$A\subset \mathbb{R}/\mathbb{Z}$.
