Local level sets of the Takagi-van der Waerden function

Abstract

In this paper, we investigate the Takagi-van der Waerden function, $$ T_r(x) = \sum_{n=0}^{\infty} \frac{ϕ(r^n x)}{r^n} ,\quad x\in [0,1], \quad r \geq 2, $$ where $ϕ(x)={\rm dist}(x,\mathbb{Z})$ represents the distance from $x$ to the nearest integer. Lagarias and Maddock[13] introduced the notion of local level sets for the classical Takagi function $T_2$. They proved that if $y$ is uniformly distributed over the range of $T_2$, then the expected number of local level sets contained in the level set at height $y$ equals $3/2$. We define the corresponding local level sets for every integer $r\geq2$ and prove that, if $y$ is uniformly distributed over the range of $T_r$, then the expected number of local level sets contained in the level set at height $y$ equals $(r+1)/r$ if $r$ is even, and $(r+1)/(r-1)$ if $r$ is odd.