A $q$-weighted analogue of the Trollope-Delange formula

TL;DR

This paper generalizes the Trollope-Delange formula by introducing a $q$-weighted version of the sequence counting binary ones, revealing different smoothness properties of associated functions depending on the value of $q$, and applying these results to ergodic fluctuations.

Contribution

It introduces a $q$-weighted analogue of the sequence and derives a new formula, connecting it to Takagi functions and ergodic theory.

Findings

For $1/2<|q|< 1$, nondifferentiable Takagi-Landsberg functions emerge.

For $|q|>1$, the functions are differentiable almost everywhere.

The results help describe fluctuations in the ergodic theorem for the dyadic odometer.

Abstract

Let $s(n)$ denote the number of "$1$"s in the dyadic representation of a positive integer $n$ and sequence $S(n) = s(1)+s(2)+\dots+s(n-1)$. The Trollope-Delange formula is a classic result that represents the sequence $S$ in terms of the Takagi function. This work extends the result by introducing a $q$-weighted analog of $s(n)$, deriving a variant of the Trollope-Delange formula for this generalization. We show that for $1/2<|q|< 1$, nondifferentiable Takagi-Landsberg functions appear, whereas for $|q|>1$, the resulting functions are differentiable almost everywhere. We further show how the result can be used to find limiting curves describing fluctuations in the ergodic theorem for the dyadic odometer.