Limiting curves for the dyadic odometer and the generalized Trollope-Delange formula

Abstract

We study limiting curves resulting from deviations in partial sums in the ergodic theorem for the dyadic odometer and non-cylindric functions. In particular, we generalize the Trollope-Delange formula for the case of the weighted sum-of-binary-digits function and show that the Takagi-Landsberg curve arises.