How large are the level sets of the Takagi function?

TL;DR

This paper investigates the size and structure of level sets of Takagi's function, revealing that while most are finite in measure, uncountably infinite level sets are residual, and many contain infinitely many local level sets.

Contribution

It provides new elementary proofs of existing results and offers a detailed description of the residual set of ordinates with uncountably infinite level sets, extending previous work.

Findings

Almost all level sets are finite in measure.

The set of ordinates with uncountably infinite level sets is residual.

Most level sets contain infinitely many local level sets.

Abstract

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.