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Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence ofintegers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve thequestion 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 theset 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 arandom set of integers known as the set of ``pseudo $s$th powers''. We considersimilar problems for subsets of the circle $\mathbb{R}/\mathbb{Z}$, that is, wepartially determine the set of $k$-tuples $(\mu(iA))_{1\leq i\leq k}$ for$A\subset \mathbb{R}/\mathbb{Z}$.

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