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

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