Let $T_{p,q}$ be line joining $(0,0)$ and $(p,q).$ Now let us define the set $$L= \bigcup_{p\in[0,1]\cap \mathbb{Q}}T_{p,1} \bigcup_{q\in[0,1]\cap \mathbb{Q}}T_{1,q} $$ and consider $P=[0,1]\times[0,1]\setminus L.$ $P$ should be a set of positive Lebesgue measure.

Question: Does there exist set of positive Lebesgue measure, $A,B \subset \mathbb{R}$ such that $A\times B\subset P?$

Addition after 1st comment:

The above property also not holds for the set $$D=\{(x,y)\in [0,1]\times [0,1]:x-y\notin \mathbb{Q}\}.$$ This is classic example and can be proved using Steinhaus theorem. But a suitable translation and rotation of $D$ contains $A\times B,$ where $A,B$ are sets of positive Lebesgue measure. This leads me to ask the above question since if $P$ doesn't have the above property, then any translation and rotation of $P$ would not also have this property.

(I believe that $P$ wouldn't contain $A\times B,$ where $A,B$ are sets of positive Lebesgue measure but I can't prove it as it doesn't have a nice definition, unlike $D$, to apply Steinhaus theorem.)