Occupation Measure and Level Sets of the Weierstrass–Cellerier Function

Summary

We show that the occupation measure of the Weierstrass–Cellerier function $\mathcal{W}(x) ={ \sum \nolimits }_{n=0}^{\infty }{2}^{-n}\sin (2\pi {2}^{n}x)$ is purely singular. Using our earlier results, we can deduce from this that almost every level set of $\mathcal{W}(x)$ is finite. These previous results and Besicovitch’s projection theorem imply that for almost every c the occupation measure of $\mathcal{W}(x,c) = \mathcal{W}(x) + cx$ is purely singular. In this chapter we verify that this result holds for all c ∈ ℝ, especially for c = 0. As happens quite often, it is not that easy to obtain from an almost everywhere true statement one that holds everywhere.