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Salem's singular function is strictly increasing, continuous, and has aderivative equal to zero almost everywhere in $[0,1]$; it is also known as deRham's singular function or Lebesgue's singular function. The parameter ofSalem's singular function $L_{\alpha}$ is $\alpha \in (0, 1)$ and $\alpha \neq1/2$. Our previous studies have shown that for some cases of which the limitset of spatio-temporal pattern of a cellular automaton (CA) is fractal, Salem'ssingular function with $\alpha = 1/3$, $1/4$, or $1/5$ is given by projectingthe pattern onto the time axis. However, it remained unclear whether thereexists a CA that gives Salem's singular function with a parameter $\alpha$equal to the multiplicative inverse of an integer greater than $5$. In thispaper, 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 impliesthat there exist CAs that give Salem's function with a parameter $\alpha$ equalto the multiplicative inverse of any integer greater than or equal to $3$. Wealso 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, thetriangular and hexagonal lattices can be considered as regular lattices in thetwo-dimensional plane, and we also discuss functions obtained from CAs on theselattices.

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