A $q$-weighted analogue of the Trollope-Delange formula
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912461364396032 |
|---|---|
| author | Minabutdinov, Aleksei |
| author_facet | Minabutdinov, Aleksei |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15201 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A $q$-weighted analogue of the Trollope-Delange formula Minabutdinov, Aleksei Number Theory Dynamical Systems 11A63 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. |
| title | A $q$-weighted analogue of the Trollope-Delange formula |
| topic | Number Theory Dynamical Systems 11A63 |
| url | https://arxiv.org/abs/2407.15201 |