Level sets of the Takagi function

Abstract: In this paper, we investigate the Takagi function, $$T_r(x) = \sum_{n=0}^{\infty} \frac{\phi(rⁿ x)}{r^n} ,\quad x\in [0,1], \quad r \in \mathbb{Z}^+,$$ where $\phi(x)={\rm dist}(x,\mathbb{Z})$ represents the distance from $x$ to the nearest integer. We generalize the concept of local level sets and find that the expected number of local level sets contained in a level set $L_r(y)$, with $y$ chosen at random, is $1 + 1/r$ for every even integer $r \geq 2$.