Mann’s theorem

Let $A$ and $B$ be subsets of $\mathbb{Z}$. If $0\in A\cap B$,

$$\sigma (A+B)\ge \min (1,\sigma A+\sigma B),$$

where $\sigma$ denotes Schnirelmann density.

This statement was known also as $(\alpha +\beta )$-conjecture until H. B. Mann proved it in 1942.