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

Abstract

It was conjectured by Furstenberg that for any $x\in [0,1]\backslash Q$, $${\dim }_H\overline{\{2^{n}x(\text{mod}1):n\ge 1\}}+{\dim }_H\overline{\{3^{n}x(\text{mod}1):n\ge 1\}}\ge 1.$$ When $x$ is a normal number, the above result holds trivially. In this note, we give explicit non-normal numbers for which the above dimensional formula holds.