2 problems

Let $F=A\times B\subset\mathbb{R}^2$, where $A$ is a set invariant under multiplication by $p$ and $B$ is a set invariant under multiplication by $q$, with integers $p,q\geq 2$ sat…

Let $p,q$ be multiplicatively independent numbers, meaning that $\frac{\log p}{\log q}$ is irrational. Let $A_p,A_q\subset[0,1]$ be respectively $p$-invariant and $q$-invariant, an…