On the distribution of the cardinalities of level sets of the Takagi function

TL;DR

This paper analyzes the distribution of the sizes of level sets of Takagi's function, revealing that most are of size two and establishing a link between the function's self-similarity and the cardinalities of its level sets.

Contribution

It introduces a set equation framework based on the self-similarity of Takagi's function to determine the distribution of level set cardinalities, including the occurrence of all positive even integers.

Findings

Most level sets of T have exactly two elements.

Any positive even integer occurs as a level set cardinality.

A simple iterative method is provided for solving T(x)=y.

Abstract

Let T be Takagi's continuous but nowhere-differentiable function. It is known that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. We show that the most common cardinality of the level sets of T is two, and investigate in detail the set of ordinates y such that the level set at level y has precisely two elements. As a by-product, we obtain a simple iterative procedure for solving the equation T(x)=y. We show further that any positive even integer occurs as the cardinality of some level set, and investigate which cardinalities occur with positive probability if an ordinate y is chosen at random from the range of T. The key to the results is a system of set equations for the level sets, which are derived from the partial self-similarity of T. These set equations yield a system of linear relationships between the cardinalities of level sets at various levels, from which all the results of this paper flow.