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.

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