Just getting around to posting thoughts I had regarding this question about the additive structure of the real numbers. I was interested in which sets generate $(\mathbb{R},+)$. First, is the following argument correct? Given any set $A$ of positive Lebesgue measure, the Steinhaus theorem says that $A-A$ contains an open neighborhood of the origin. As per Arturo Magidin's answer to the original question, any such interval generates $\mathbb{R}$. Noting that $A-A$ is contained in the subgroup of $\mathbb{R}$ generated by $A$, we see that $A$ in fact generates $\mathbb{R}$.
Second, are there sets of measure zero which generate $\mathbb{R}$? I looked around a bit, but am not really sure what tools to use to approach this question.