If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$.

I would like to see an example of such a function with a NON $\sigma$ finite graph (with respect to $\mathscr{H}^{1}$).

  • 1
    $\begingroup$ Do you know such examples exist or are you implicitly asking whether or not they do? $\endgroup$ – Thompson Jun 5 '17 at 23:45
  • $\begingroup$ I do not know if such example exists. However I would be surprised if it does not, because in several papers I have seen non trivial proofs for the $\sigma$ finiteness of the graphs of functions in special subclasses of the class $\bigcap_{0 < \alpha < 1}\mathscr{C}^{0,\alpha}$, everyone using some special feature of the subclass. $\endgroup$ – Longyearbyen Jun 6 '17 at 8:04
  • $\begingroup$ Did you find the answer? I am sure there is an example with a non $\sigma$-finite 1-measure. $\endgroup$ – Piotr Hajlasz May 17 '20 at 18:34

I believe the Takakgi function satisfies this property. According to The Takagi Function: A Survey, the Takagi function $T$ satisfies $$T(x+h)-T(x) = O(h\log(1/|h|) \: \text{ as } \: h\to0$$ and this is the best possible estimate.

Of course, lower bounds for Hausdorff measure are tricky but I think the result you need can be found in the paper On the Hausdorff Dimension of Some Graphs by Mauldin and Williams.

  • $\begingroup$ Thank you for your answer. In this survey they claim that the graph is $\sigma$ finite, see p.16. Actually the fact that the takagi function has $\sigma$ finite graph was the initial motivation for my question. $\endgroup$ – Longyearbyen Jun 5 '17 at 22:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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