The Hausdorff dimension of the nondifferentiability set of the Cantor function is [𝑙𝑛(2)/𝑙𝑛(3)

Abstract

The main purpose of this note is to verify that the Hausdorff dimension of the set of points N ∗ {N^{\ast }} at which the Cantor function is not differentiable is [ ln ⁡ ( 2 ) / ln ⁡ ( 3 ) ] 2 {[\ln (2)/\ln (3)]^2} . It is also shown that the image of N ∗ {N^{\ast }} under the Cantor function has Hausdorff dimension ln ⁡ ( 2 ) / ln ⁡ ( 3 ) \ln (2)/\ln (3) . Similar results follow for a standard class of Cantor sets of positive measure and their corresponding Cantor functions.