Level Sets of the Takagi Function: Local Level Sets

TL;DR

This paper investigates the structure of level sets of the Takagi function, revealing that for most points, the level sets are uncountably infinite, and introduces local level sets to analyze their properties.

Contribution

It introduces the concept of local level sets for the Takagi function and analyzes their structure, showing that typical level sets are uncountably infinite and the expected number of local level sets at a random level is 1.5.

Findings

For a generic set of points, level sets are uncountably infinite.

The expected number of local level sets at a random level is 1.5.

A singular monotone function related to local level sets is constructed.

Abstract

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a "generic" full Lebesgue measure set of ordinates y, the level sets are finite sets. Here it is shown for a "generic" full Lebesgue measure set of abscissas x, the level set L(\tau(x)) is uncountable. An interesting singular monotone function is constructed, associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly 3/2.