Let $\Omega =\{0,1\}^{\mathbb{N}}$ denote the set of infinite sequences with elements $0$ or $1$. Let $d$ be the metric on $\Omega$ given by $d((x\_n),(y\_n))=1/2^m$, where $m=\min\{i\in\mathbb{N}\,:\,x\_i\neq y\_i\}$.


Define a function $\pi:\Omega\rightarrow [0,1]$, where $\pi((x\_n))$ is the number in $[0,1]$ with $n$th digit in its binary expansion equal to $x\_n$, that is, $\pi ((x\_n)) = \sum\limits\_{n=1}^{\infty}x\_n2^{-n}$.


Does anyone have a *reference* for published work which proves that the Hausdorff dimension of a set $X$ in the metric space $(\Omega,d)$ is equal to the Hausdorff dimension of $\pi(X)$ in $[0,1]$ with the Euclidean metric? 



Hausdorff dimension of sequence space
According to Falconer[1] this is due to Besicovitch[2]. Falconer states it (generalized to $\mathbb R^n$) as Theorem 5.1, p. 65. This proves more than just $X$ and $\pi(X)$ have the same Hausdorff dimension: it proves the Hausdorff measures of $X$ and $\pi(X)$ are within a constant factor of each other.


[1] K. F. Falconer, *The Geometry of Fractal Sets* (Cambridge Univ Press 1985), Chapter 5


[2] A. S. Besicovitch, *Indag. Math.* **14** (1952) 339-344


