Local level sets of the Takagi-van der Waerden function

In this paper, we investigate the Takagi-van der Waerden function, $$T_r(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{\phi (r^{n}x)}{r^n},x\in [0,1],r\ge 2,$$ where $\phi (x)=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathbb{Z})$ 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 $3\mathrm{/}2$. We define the corresponding local level sets for every integer $r\ge 2$ 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 $(r+1)\mathrm{/}r$ if $r$ is even, and $(r+1)\mathrm{/}(r-1)$ if $r$ is odd.