论文标题
在与模式序列相关的二进制单词的单色算术上
On monochromatic arithmetic progressions in binary words associated with pattern sequences
论文作者
论文摘要
令$ e_v（n）$表示$ n \ in \ mathbb {n} $的二进制扩展中固定模式$ v $的出现数量。在本文中，我们研究了二进制单词类别的单色算术进程$（e_v（n）\ bmod {2}）_ {n \ geq 0} $，其中包括著名的thue-morse word $ \ mathbf {t} $ \ mathbf {t} $和rudin-shapiro word $ \ shapiro word $ \ m m iathbf} $。我们证明，单色算术差异的差异$ d \ geq 3 $的长度最多是$ \ m mathbf {r} $，最多是$（d+3）/2 $，对于无限的$ d $而言，均等。此外，我们计算了$ \ mathbf {r} $差异$ 2^k-1 $和$ 2^k+1 $的单色算术进程的最大长度。对于一般模式$ v $，我们提供了任何差异$ d $的单色算术进程的上限。我们还证明了其他其他结果，并提供了许多相关问题和猜想。
Let $e_v(n)$ denote the number of occurrences of a fixed pattern $v$ in the binary expansion of $n \in \mathbb{N}$. In this paper we study monochromatic arithmetic progressions in the class of binary words $(e_v(n) \bmod{2})_{n \geq 0}$, which includes the famous Thue--Morse word $\mathbf{t}$ and Rudin--Shapiro word $\mathbf{r}$. We prove that the length of a monochromatic arithmetic progression of difference $d \geq 3$ starting at $0$ in $\mathbf{r}$ is at most $(d+3)/2$, with equality for infinitely many $d$. Moreover, we compute the maximal length of a monochromatic arithmetic progression in $\mathbf{r}$ of difference $2^k-1$ and $2^k+1$. For a general pattern $v$ we provide an upper bound on the length of a monochromatic arithmetic progression of any difference $d$. We also prove other miscellaneous results and offer a number of related problems and conjectures.
