Are there sets A and B such that any $x \in \mathbb{R}$ can be decomposed as $x = a+b$, $a \in A$ and $b \in B$, where the Lebesgue measure of $A$ and of $B$ is null.
There is an indication that this should follow from the fact that $C_q = \{z \in[0,1]; z_i\in\{0,2,\cdots,q-1\}\}$ has null measure for every $q \in \mathbb{N}$, where $z_i$ are the numbers in the q-adic expansion of $z$, that is $$z=\sum_{i=1}^{\infty}\frac{z_i}{q^i}$$