$D$-dimensional cellular automata provide Salem's singular function $L_{\alpha}$ with $\alpha=1/(2D+1)$ and $1/(2^D+1)$

TL;DR

This paper constructs cellular automata in various dimensions that generate Salem's singular function with specific parameters, expanding understanding of the connection between CA patterns and fractal functions.

Contribution

It introduces new cellular automata models that produce Salem's singular function with parameters as reciprocals of integers greater than 2, for multiple lattice types and dimensions.

Findings

Constructed CAs for /(2D+1) and /(2^D+1) parameters in all dimensions D .

Numerical experiments show limitations for certain CA types in generating Salem's functions with /M parameters.

Extended analysis to triangular and hexagonal lattices in addition to square lattices.

Abstract

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 $L_{\alpha}$ is $\alpha \in (0, 1)$ and $\alpha \neq 1/2$. 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 $\alpha = 1/3$, $1/4$, 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 $\alpha$ equal to the multiplicative inverse of an integer greater than $5$. In this paper, we construct CAs giving Salem's singular function with $\alpha= 1/(2D+1)$ and $\alpha = 1/(2^D+1)$ for each dimension $D \geq 1$. This implies that there exist CAs that give Salem's function with a parameter $\alpha$ 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 $\alpha = 1/M$ 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.