Two disjoint sets of finite measure that are closed under summation and multiplication

Question

I am looking for two subsets of $\mathbb{R}$, which we denote by $A,B$ with positive and finite measure (Lebesgue measure) that intersect only at ${\mathbb{Q}}$. Moreover, I want that both $A$ and $B$ will be closed under summation, taking negative, multiplication and taking inverse.

If anyone saw such sets, let me know.

Thanks.

Answer 1

By Steinhaus theorem both $A$ and $B$ would contain open neighbourhood of the origin.