On the level sets of the Takagi-van der Waerden functions

This paper examines the level sets of the continuous but nowhere differentiable functions \begin{equation*} f_r(x)=\sum_{n=0}^\infty r^{-n}\phi(r^n x), \end{equation*} where $\phi(x)$ is the distance from $x$ to the nearest integer, and $r$ is an integer with $r\geq 2$. It is shown, by using properties of a symmetric correlated random walk, that almost all level sets of $f_r$ are finite (with respect to Lebesgue measure on the range of $f$), but that for an abscissa $x$ chosen at random from $[0,1)$, the level set at level $y=f_r(x)$ is uncountable almost surely. As a result, the occupation measure of $f_r$ is singular.