Local level sets of the Takagi-van der Waerden function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912895478005760 |
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| author | Jiang, Lai Ying, Ting-Ting Zhang, Yi-Yang |
| author_facet | Jiang, Lai Ying, Ting-Ting Zhang, Yi-Yang |
| contents | 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 $1 + 1/r$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_21683 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local level sets of the Takagi-van der Waerden function Jiang, Lai Ying, Ting-Ting Zhang, Yi-Yang Classical Analysis and ODEs 28A80, 41A30 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 $1 + 1/r$. |
| title | Local level sets of the Takagi-van der Waerden function |
| topic | Classical Analysis and ODEs 28A80, 41A30 |
| url | https://arxiv.org/abs/2508.21683 |