Let *σ* be a constant in the interval (0, 1), and let *A* be an infinite set of positive integers which contains at least *c*_{1}*x*^{σ} and at most *c*_{2}*x*^{σ} elements in the interval [1, *x*] for some constants *c*_{2} > *c*_{1} > 0 independent of *x* and each *x* ≥ *x*_{0}. We prove that then the sumset *A + A* has more elements than *A* (counted up to *x*) by a factor