Level Sets of the Takagi Function: Generic Level Sets

Details Level Sets of the Takagi Function: Generic Level Sets Level Sets of the Takagi Function: Generic Level Sets

2010-11-14

arxiv.org/abs/1011.3183v4

Mathematics

Classical Analysis and ODEs

Comments welcome. 23 pages, 2 figures. Latest version is an extensive rewrite of earlier versions

Scientific paper

The Takagi function {\tau} : [0, 1] \rightarrow [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. This paper studies the level sets L(y) = {x : {\tau}(x) = y} of the Takagi function {\tau}(x). It shows that for a full Lebesgue measure set of ordinates y, these level sets are finite sets, but whose expected number of points is infinite. Complementing this, it shows that the set of ordinates y whose level set has positive Hausdorff dimension is itself a set of full Hausdorff dimension 1 (but Lebesgue measure zero). Finally it shows that the level sets have a nontrivial Hausdorff dimension spectrum. The results are obtained using a notion of "local level set" introduced in a previous paper, along with a singular measure parameterizing such sets.

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