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?