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

TL;DR

This paper constructs explicit non-normal numbers for which the sum of the Hausdorff dimensions of the orbits under multiplication by 2 and 3 exceeds or equals 1, addressing a conjecture related to Furstenberg's problem.

Contribution

It provides explicit examples of non-normal numbers satisfying Furstenberg's dimension conjecture, expanding understanding beyond normal numbers.

Findings

Explicit non-normal numbers satisfying the conjecture.

Validation of the dimension sum for specific non-normal points.

Addresses a special case of Furstenberg's conjecture.

Abstract

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.