论文标题
$ d $ - 维蜂窝自动机提供塞勒姆的单数函数$l_α$，$α= 1/（2d+1）$和$ 1/（2^d+1）$
$D$-dimensional cellular automata provide Salem's singular function $L_α$ with $α=1/(2D+1)$ and $1/(2^D+1)$
论文作者
论文摘要
塞勒姆的奇异功能严格增加，连续，并且在$ [0,1] $中几乎到处都等于零。它也被称为De Rham的单数功能或Lebesgue的单数功能。塞勒姆（Salem）单数函数$l_α$的参数为$α\ in（0，1）$和$α\ neq 1/2 $。我们先前的研究表明，在某些情况下，细胞自动机（CA）的时空模式的极限集为分形，塞勒姆的奇异功能具有$α= 1/3 $，$ 1/4 $或$ 1/5 $，是通过将图案投影到时间轴上的。但是，目前尚不清楚是否存在一个CA，该CA具有塞勒姆的奇异功能，其参数$α$等于整数大于$ 5 $的乘法倒数。在本文中，我们构建了CAS，以$α= 1/（2d+1）$和$α= 1/（2^d+1）$为每个维度$ d \ geq 1 $。这意味着存在CAS的CAS，可以使Salem的功能具有参数$α$等于任何大于或等于$ 3 $的乘法倒数。我们还提出了数值实验的结果表明，对于$ d \ leq 5 $，由$ d $ d $ - 二维线性对称对称$ 2 $ 2 $ - 状态半径 - $ 1 $ 1 $ 1 $ cas除了上述两种类型以外，不能是$α= 1/m $ for $ m $ for $ m $ for $ m \ in {\ Mathbb z} \ geq 3}。除方形晶格外，三角形和六角形晶格可以被视为二维平面中的常规晶格，我们还讨论了从CAS上获得的这些晶格的功能。
Salem's singular function is strictly increasing, continuous, and has a derivative equal to zero almost everywhere in $[0,1]$; it is also known as de Rham's singular function or Lebesgue's singular function. The parameter of Salem's singular function $L_α$ is $α\in (0, 1)$ and $α\neq 1/2$. Our previous studies have shown that for some cases of which the limit set of spatio-temporal pattern of a cellular automaton (CA) is fractal, Salem's singular function with $α= 1/3$, $1/4$, or $1/5$ is given by projecting the pattern onto the time axis. However, it remained unclear whether there exists a CA that gives Salem's singular function with a parameter $α$ equal to the multiplicative inverse of an integer greater than $5$. In this paper, we construct CAs giving Salem's singular function with $α= 1/(2D+1)$ and $α= 1/(2^D+1)$ for each dimension $D \geq 1$. This implies that there exist CAs that give Salem's function with a parameter $α$ equal to the multiplicative inverse of any integer greater than or equal to $3$. We also present the results of numerical experiments showing that for $D \leq 5$, the functions given by $D$-dimensional linear symmetric $2$-state radius-$1$ CAs other than the above two types cannot be Salem's function with $α= 1/M$ for $M \in {\mathbb Z}_{\geq 3}$. In addition to the square lattice, the triangular and hexagonal lattices can be considered as regular lattices in the two-dimensional plane, and we also discuss functions obtained from CAs on these lattices.
