Cellular Automata and Powers of $p/q$

TL;DR

This paper investigates the behavior of cellular automata that multiply numbers by p/q in base pq, revealing conditions under which fractional parts of these sequences are densely contained in unions of intervals.

Contribution

It introduces a novel analysis of cellular automata $F_{p,q}$ related to multiplication by $p/q$, establishing new results on the distribution of fractional parts of these sequences.

Findings

For $p \,\geq\, 2q-1$, fractional parts are contained in small unions of intervals.

For $p > q > 1$, certain unions of intervals approximate the unit interval but exclude all fractional parts.

The study links cellular automata dynamics with number theory and measure theory.

Abstract

We consider one-dimensional cellular automata $F_{p,q}$ which multiply numbers by $p/q$ in base $pq$ for relatively prime integers $p$ and $q$. By studying the structure of traces with respect to $F_{p,q}$ we show that for $p\geq 2q-1$ (and then as a simple corollary for $p>q>1$) there are arbitrarily small finite unions of intervals which contain the fractional parts of the sequence $\xi(p/q)^n$, ($n=0,1,2,\dots$) for some $\xi>0$. To the other direction, by studying the measure theoretical properties of $F_{p,q}$, we show that for $p>q>1$ there are finite unions of intervals approximating the unit interval arbitrarily well which don't contain the fractional parts of the whole sequence $\xi(p/q)^n$ for any $\xi>0$.