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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?

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2 Answers 2

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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

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  • $\begingroup$ are you sure that link [2] is correct? couldn't find it on google scholar $\endgroup$
    – Amir Sagiv
    Commented Apr 20, 2016 at 15:31
  • $\begingroup$ It seems to be Indag. Math. not Invent. Math ... corrected ... Besicovitch, A. S. On existence of subsets of finite measure of sets of infinite measure. Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math. 14, (1952). 339–344. $\endgroup$
    – Gerald Edgar
    Commented Apr 20, 2016 at 15:36
  • $\begingroup$ Thank you Gerald, that was extremely useful. I now see that the material you reference can also be found, in brief, in Section 2.4 of another of Falconer's books, Fractal Geometry. $\endgroup$
    – Ian Short
    Commented Apr 21, 2016 at 9:09
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This observation is attributed to H. Furstenberg, and appears (in the case of shift-invariant sets, i.e. Cantor sets) in his beautiful Disjointness paper (in section $3$, which you can read independently from the previous ones, although the whole paper is magnificent). A bit more general result appears in a subsequent paper of Furstenberg named "Intersections of Cantor sets"

I'm pretty sure such a result was known much before Furstenberg's (at-least to Erdos) but you wanted some specific references.

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  • $\begingroup$ Thanks very much Amir! I'm extremely grateful for the reference - to solve my problem, and also to alert me to another outstanding paper of Furstenburg. $\endgroup$
    – Ian Short
    Commented Apr 21, 2016 at 9:13

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