Level Sets of the Takagi Function: Generic Level Sets

Abstract

The Takagi function τ : [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 : τ(x) = y} of the Takagi function τ(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.

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

References in corpus (4)

• Level Sets of the Takagi Function: Local Level Sets
• The Takagi Function and Its Properties
• The improper infinite derivatives of Takagi's nowhere-differentiable function
• On the distribution of the cardinalities of level sets of the Takagi function

Cited by in corpus (6)

• Level Sets of the Takagi Function: Local Level Sets
• Slices of the Takagi function
• The Takagi function: a survey
• How large are the level sets of the Takagi function?
• On the distribution of the cardinalities of level sets of the Takagi function
• Hausdorff dimension of level sets of generalized Takagi functions