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

This paper examines the level sets of the continuous but nowhere differentiable functions f r ( x ) = ∑ n = 0 ∞ r − n ϕ ( r n x ) , 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 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.