Note on the points with dense orbit under $\times 2$ and $\times 3$ maps

Wenya Wang

It was conjectured by Furstenberg that for any $x\in [0,1]\backslash Q$, $$ \dim_H \bar{\{2^nx ({\text{mod}}\ 1): n\ge 1\}}+ \dim_H \bar{\{3^nx ({\text{mod}}\ 1): n\ge 1\}}\ge 1. $…