8
$\begingroup$

I use $\dim_H(E)$ to denote the Hausdorff dimension of a set $E \subseteq \mathbb{R}$ and $|E|$ to denote its Lebesgue measure. It is easy to see from the definition of Hausdorff dimension that if $\dim_H(E) < 1$, then $|E| = 0$. The converse is not true, and there are many cases where $\dim_H(E) = 1$ yet $|E| = 0$. So the question:

What was the first (or most elementary) example of this phenomenon?

After some looking around, I was able to prove that a central Cantor set $C$ with ratio of dissection $r_k = 1/(2+\frac{1}{k})$ satisfies the condition I want. It is easy to see that $|C| = 0$ since at step $n$ of the process to construct this Cantor set, it has measure $2^n(r_1 \cdots r_n)$ which in this case limits to 0, but for the Hausdorff dimension I required a non-trivial result from the paper Sums of Cantor sets (Cabrelli, Hare, Molter) that gave the formula

$\dim_H(C) = \liminf_n \frac{n \ln 2}{\ln r_1 \cdots r_n}$.

This result is fairly recent and sophisticated, and I feel that there should be older and simpler examples.

$\endgroup$

4 Answers 4

16
$\begingroup$

Try a countable union of sets (such as Cantor sets) whose Hausdorff dimension tends to 1.

$\endgroup$
1
  • $\begingroup$ Thank you! I was mired in the idea that individual Cantor sets can never have Hausdorff dimension 1, but of course one can appeal to countable stability. $\endgroup$
    – Vince
    Aug 18, 2010 at 20:13
4
$\begingroup$

Perhaps Hausdorff's original paper? He uses gauge functions other than powers of x. And constructs Cantor sets corresponding to them. For example if you take $x |\log x|$ then you get a set of Hausdorff dimension $1$ but measure $0$.

$\endgroup$
3
$\begingroup$

It is a very common phenomenon in ergodic theory when the set of points which do not satisfy the Birkhoff ergodic theorem (i.e., a set of zero measure) has full Hausdorff dimension.

See, for instance, http://www.math.psu.edu/pesin/papers_www/birk.pdf

$\endgroup$
1
$\begingroup$

Perhaps the following charts "Universal measure zero sets with full Hausdorff dimension" http://mat.fsv.cvut.cz/Zindulka/papers/opava.pdf are of interest for you.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.