Decomposing the sum-of-digits correlation measure
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910926913929216 |
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| author | Sobolewski, Bartosz Spiegelhofer, Lukas |
| author_facet | Sobolewski, Bartosz Spiegelhofer, Lukas |
| contents | Let $s(n)$ denote the number of ones in the binary expansion of the nonnegative integer $n$. How does $s$ behave under addition of a constant $t$? In order to study the differences \[s(n+t)-s(n),\] for all $n\ge0$, we consider the associated characteristic function $γ_t$. Our main theorem is a structural result on the decomposition of $γ_t$ into a sum of \emph{components}. We also study in detail the case that $t$ contains at most two blocks of consecutive $1$s. The results in this paper are motivated by \emph{Cusick's conjecture} on the sum-of-digits function. This conjecture is concerned with the \emph{central tendency} of the corresponding probability distributions, and is still unsolved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07779 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decomposing the sum-of-digits correlation measure Sobolewski, Bartosz Spiegelhofer, Lukas Number Theory Primary: 11B65, 11A63, Secondary: 05A20, 11T71 Let $s(n)$ denote the number of ones in the binary expansion of the nonnegative integer $n$. How does $s$ behave under addition of a constant $t$? In order to study the differences \[s(n+t)-s(n),\] for all $n\ge0$, we consider the associated characteristic function $γ_t$. Our main theorem is a structural result on the decomposition of $γ_t$ into a sum of \emph{components}. We also study in detail the case that $t$ contains at most two blocks of consecutive $1$s. The results in this paper are motivated by \emph{Cusick's conjecture} on the sum-of-digits function. This conjecture is concerned with the \emph{central tendency} of the corresponding probability distributions, and is still unsolved. |
| title | Decomposing the sum-of-digits correlation measure |
| topic | Number Theory Primary: 11B65, 11A63, Secondary: 05A20, 11T71 |
| url | https://arxiv.org/abs/2411.07779 |