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

This paper examines the level sets of the continuous but nowhere differentiable functionsfr(x)=∑n=0∞r−nϕ(rnx), where ϕ(x) is the distance from x to the nearest integer, and r is an integer with r≥2. It is shown, by using properties of a symmetric correlated random walk, that almost all level sets of fr 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=fr(x) is uncountable almost surely. As a result, the occupation measure of fr is singular.