Let $P\subset \mathbb{R}^2$ be a set of positive Lebesgue measure. Is it always true that a suitable rotation and translation of $P$ always contains a set of the form $\{re^{i\theta}:r\in E, \theta\in [0,2\pi)\}$ or $A×B,$ where $E,A,B$ are sets of positive Lebesgue measure in $\mathbb{R}?$

Note: I can show that the two types of positive measure sets, mentioned above, are different (in the sense that no one type contains the other type always).