تقرير
Block occurrences in the binary expansion
| العنوان: | Block occurrences in the binary expansion |
|---|---|
| المؤلفون: | Sobolewski, Bartosz, Spiegelhofer, Lukas |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, 11A63, 05A20 (Primary) 05A16, 11T71 (Secondary) |
| الوصف: | The binary sum-of-digits function $\mathsf{s}$ returns the number of ones in the binary expansion of a nonnegative integer. Cusick's Hamming weight conjecture states that, for all integers $t\geq 0$, the set of nonnegative integers $n$ such that $\mathsf{s}(n+t)\geq \mathsf{s}(n)$ has asymptotic density strictly larger than $1/2$. We are concerned with the block-additive function $\mathsf{r}$ returning the number of (overlapping) occurrences of the block $\mathtt{11}$ in the binary expansion of $n$. The main result of this paper is a central limit-type theorem for the difference $\mathsf{r}(n+t)-\mathsf{r}(n)$: the corresponding probability function is uniformly close to a Gaussian, where the uniform error tends to $0$ as the number of blocks of ones in the binary expansion of $t$ tends to $\infty$. Comment: 19 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2309.00142 |
| رقم الأكسشن: | edsarx.2309.00142 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |