Cellular automata that generate symmetrical patterns give singular functions

Discontinuous Riemann integrable functions emerging from cellular automata

“…In [6,7], we showed several examples, whose one-variable functions are Salem's singular function [10,1,8,14], that is a kind of strictly increasing singular functions. In [4], we showed the existence of D-dimensional symmetrical CAs that give one-variable functions such that the parameters of Salem's singular function are 1/(2D + 1) and 1/(2 D + 1), and in [5], we showed that the one-variable function obtained for one-dimensional elementary CA Rule 150 is a new singular function other than Salem's singular function. In [3], sufficient conditions are given for a function obtained from a CA to be a singular function.…”