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

TL;DR

This paper investigates the limiting behavior of partial sum deviations in the dyadic odometer, generalizing the Trollope-Delange formula and linking it to the Takagi-Landsberg curve, revealing new fractal structures in ergodic sums.

Contribution

It extends the Trollope-Delange formula to weighted binary digit sums and connects these deviations to the Takagi-Landsberg curve, broadening understanding of ergodic sum fluctuations.

Findings

Generalized Trollope-Delange formula for weighted sums

Identification of Takagi-Landsberg curve as a limiting shape

Analysis of deviations in non-cylindric functions

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.