Local level sets of the Takagi-van der Waerden function

TL;DR

This work extends the study of level sets from the classical Takagi function to the Takagi-van der Waerden family $T_r$ with even $r$, establishing a precise probabilistic description of local level sets. The authors define an $r$-based equivalence $\sim_r$ to form local level sets and develop the geometry of humps as scaled copies of the full graph, enabling a measure-theoretic enumeration of local level sets within level sets. The main result shows that for a uniformly distributed $y$ on the range of $T_r$, the expected number of local level sets contained in $L_r(y)$ equals $1+\frac{1}{r}$, while the expected size of $L_r(y)$ is infinite; surprisingly, level sets are finite for almost every $y$, but the average cardinality diverges due to contributions from humps. These findings generalize Lagarias–Maddock's r=2 case and connect the local structure of level sets to hump geometry and fractal-like counting.

Abstract

In this paper, we investigate the Takagi-van der Waerden function, $$ T_r(x) = \sum_{n=0}^{\infty} \frac{φ(r^n x)}{r^n} ,\quad x\in [0,1], \quad r \in \mathbb{Z}^+, $$ where $φ(x)={\rm dist}(x,\mathbb{Z})$ represents the distance from $x$ to the nearest integer. %We prove that for every even integer $r \geq 2$, the expected number of local level sets contained in the level set $L_r(y)$ is $1 + 1/r$, if $y$ is a random variable uniformly distributed over the range of $T_r$. Lagarias and Maddock [Level sets of the Takagi function: local level sets, \emph{Monatsh. Math.}, {\bf 166} (2012), No. 2, 201--238] introduced the notion of local level sets for the classical Takagi function $T_2$. They proved that if $y$ is a random variable uniformly distributed over the range of $T_2$, then the expected number of local level sets contained in the level set $L_2(y)$ equals $3/2$. We extend the study by defining an analogous concept of local level sets for all even integers $r$. Then we prove that, for every even integer $r\geq 2$, if $y$ is a random variable uniformly distributed, then the expected number of local level sets contained in the level set $L_r(y)$ equals $1 + 1/r$.

Figures (1)

• Figure 1: Let $r=4$. The left picture shows all the humps of order $1$. The right picture displays the humps close to the hump $H(3/16)$. The red humps are leading humps with generation $1$, while the orange ones are leading humps with generation $2$.

Theorems & Definitions (29)

• Theorem 1.1: LM12
• Theorem 1.2
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5