Cellular automata and powers of p/q

Abstract

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