Irregular \(1\)-sets on the graphs of continuous functions
The author considers the Takagi function on \([0,1]\) given by \[ \mathcal{T}(x)=\sum_{n=0}^\infty2^{-n}| | 2^nx| | , \] where \(| | x| | \) is the distance of \(x\) from the closest integer. In Theorem 9, it is proved that the occupation measure of a Takagi function is singular and its graph can be decomposed into two sets: an irregular 1-set \(\mathcal{S}_{\text{irr}}\) and a set \(\mathcal {S}_{\text{reg}}\) which can be covered by the graphs of countably many increasing functions. Further, the Lebesgue measure of the projection on \(Ox\) of \(\mathcal{S}_{\text{irr}}\) is \(1\) while the Lebesgue measure of its projection on \(Oy\) is \(0\). In section 5 of this paper, a class of ``perturbed'' Weierstrass-Cellerier-type functions is considered. In Theorem 13, it is proved that the set \(\mathcal{S}_{\text{irr}}\) is an irregular 1-set, the set \(\mathcal{S}_{\text{reg}}\) can be covered by the union of the graphs of countably many strictly monotone functions and by a set of zero 1-Hausdorff measures. The paper is very well written and the subject is interesting and up to date.
scientific article; zbMATH DE number 5651925