On the structure of sequences with minimal maximal pattern complexity
Casey Schlortt
TL;DR
This paper analyzes low maximal pattern complexity $p_{\alpha}^*(k)$ for aperiodic sequences on alphabets of size $\ell\ge3$. Using singular decomposition, residue analyses, and auxiliary words, it proves a general structure theorem: under $\liminf_{k\to\infty} (p_{\alpha}^*(k)-3k)=-\infty$, exactly one residue is aperiodic over two letters and all other residues are constant (uniform recurrence improves the bound to $4k$). It then fully characterizes strong pattern Sturmian sequences, showing they are pattern Sturmian on two letters threaded with constants, and proves a partial converse linking residue decompositions to $p_{\alpha}^*(k)=2k+C$ with $C\ge\ell-2$; equality $C=\ell-2$ yields strong pattern Sturmian behavior. The paper concludes with explicit constructions achieving the minimal bound $p_{\alpha}^*(k)=2k+\ell-2$ and examples illustrating bound sharpness and the limits of the converses.
Abstract
In 2002, Kamae and Zamboni introduced maximal pattern complexity and determined that any aperiodic sequence must have maximal pattern complexity at least $2k$. In 2006, Kamae and Rao examined the maximal pattern complexity of sequences over larger alphabets and showed that sequences which have maximal pattern complexity less than $\ell k$, for $\ell$ the size of the alphabet, must have some periodic structure. In this paper, we investigate the structure of sequences of low maximal pattern complexity over $\ell$ letters where $\liminf\limits\limits_{k \to \infty} p_α^*(k) - 3k = -\infty$. In addition, we show that the minimal maximal pattern complexity of an aperiodic sequence which uses all $\ell$ letters is $p_α^*(k) = 2k + \ell -2$, and give an exact structure for aperiodic sequences with this maximal pattern complexity.