It can be shown that the pointwise sum of two measure zero sets is not necessary of measure zero, take for example the Canter set $C$, we have $C+C=[0,2]$.
Now my question is, what about the pointwise multiplication of two measure zero sets in real numbers, in the sense that $A\cdot B=\left\{a\cdot b: a\in A,~b\in B \right\}$.
Consider the set $B=e^{C} =\{e^x :x\in C\}.$ Then you have $$B\cdot B =e^C \cdot e^C =e^{C+C} .$$