Level sets of signed Takagi functions

TL;DR

This paper studies the structure of level sets of signed Takagi functions, revealing that most are finite, but uncountably large level sets are topologically residual, and extends the concept of local level sets to these functions.

Contribution

It extends the analysis of Takagi functions by characterizing level sets of signed variants and relating their size to the function's height, introducing new insights into their structure.

Findings

Almost all level sets are finite in Lebesgue measure.

Uncountably large level sets form a residual set in the range.

Average number of local level sets per level is between 1.5 and 2.

Abstract

This paper examines level sets of functions of the form $f(x)=\sum_{n=0}^\infty \frac{r_n}{2^n}\phi(2^n x)$, where phi(x) is the distance from x to the nearest integer, and r_n equals 1 or -1 for each n. Such functions are referred to as signed Takagi functions. The case when r_n=1 for all n is the classical Takagi function, a well-known example of a continuous but nowhere differentiable function. For f of the above form, the maximum and minimum values of f are expressed in terms of the sequence {r_n}. It is then shown that almost all level sets of f are finite (with respect to Lebesgue measure on the range of f), but the set of ordinates y with an uncountably large level set is residual in the range of f. The concept of a local level set of the Takagi function, due to Lagarias and Maddock, is extended to arbitrary signed Takagi functions. It is shown that the average number of local level sets contained in a level set of f is the reciprocal of the height of the graph of f, and consequently, this average lies between 3/2 and 2.