Local level sets of the Takagi-van der Waerden function

In this paper, we investigate the Takagi-van der Waerden function, $$T_r(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{\phi (r^{n}x)}{r^n},x\in [0,1],r\in {\mathbb{Z}}^{+},$$ where $\phi (x)=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathbb{Z})$ represents the distance from $x$ to the nearest integer. %We prove that for every even integer $r\ge 2$, the expected number of local level sets contained in the level set $L_r(y)$ is $1+1\mathrm{/}r$, 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 $3\mathrm{/}2$. 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 $r\ge 2$, 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 $1+1\mathrm{/}r$.