Level Sets of the Takagi Function: Generic Level Sets

TL;DR

This paper investigates the structure of level sets of the Takagi function, revealing that most have finite points with infinite expected size, while the set of ordinates with complex level sets is itself highly intricate.

Contribution

It characterizes the measure and Hausdorff dimension properties of the level sets of the Takagi function, introducing new tools like local level sets and singular measures.

Findings

Most level sets are finite but have infinite expected size.

The set of ordinates with positive Hausdorff dimension has full Hausdorff dimension 1.

Level sets exhibit a nontrivial Hausdorff dimension spectrum.

Abstract

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.