Does a given positive measure set in $\mathbb{R}^2$ contains product of two 1 dimensional positive measure set? In $\mathbb{R}^2$ consider the 2 dimensional positive Lebesgue measure set,
$$D=\{re^{it}:t\in[0,2\pi),r\in[0,1]\setminus\mathbb{Q}\}  .   $$
Does there exist sets $A,B\subset\mathbb{R}$ of 1 dimensional positive  Lebesgue measure such that $$A\times B\subset D?$$
Note: The above is not true if $$D=\{(x,y)\in [0,1]\times [0,1]:x-y\notin\mathbb{Q}\}.$$

No, it there were two such sets $A$, $B$,  then with the map $(x,y) \mapsto (x^2, y^2)$ ($t \mapsto t^2$ bi-Lipschitz on compacts inside $(0, \infty)$) , it would transform into $A_1$, $B_1$ of measure non-zero inside $\{(x,y)\ | \ x+y \not \in \mathbb{Q}^2\}$, which is not possible.
