Sumsets of measurable sets

Abstract:

Let ${\mathcal {A}_1},{\mathcal {A}_2}, \ldots ,{\mathcal {A}_n}$ be Lebesgue measurable sets of positive real numbers such that ${\inf \mathcal {A}_i} = 0$ for all i. Let $\mu$ denote Lebesgue measure and let ${\mu _ \ast }$ denote inner Lebesgue measure. If $\sum \nolimits _{i = 1}^n \mu ({\mathcal {A}_i} \cap [0,t]) \geqslant \gamma t$ for some $\gamma \leqslant 1$ and all $t \leqslant x$, then \[ {\mu _ \ast }(({\mathcal {A}_1} + {\mathcal {A}_2} + \cdots + {\mathcal {A}_n}) \cap [0,x]) \geqslant \gamma x.\] This generalizes results of Dyson and Macbeath.