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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.15201 |
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Table of Contents:
- 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.