Abelian Nivat's conjecture for non-rectangular patterns

TL;DR

This paper investigates the relationship between abelian pattern complexity and periodicity in two-dimensional words, establishing that constant abelian complexity implies full periodicity for convex patterns, and explores related properties in one dimension.

Contribution

It characterizes convex patterns in two dimensions where abelian complexity equals one, proving such words are necessarily fully periodic, and extends analysis to functions on f2 with constant sums.

Findings

Constant abelian pattern complexity implies full periodicity in 2D convex patterns.

Similar results hold for functions on f2 with constant sums.

Characterization of patterns allowing non-constant functions with constant sums in 1D.

Abstract

In this paper, we study the relation between periodicity of two-dimensional words and their abelian pattern complexity. A pattern $\cal{P}$ in $\mathbb{Z}^n$ is the set of all translations of some finite subset $F$ of $\mathbb{Z}^n$. An $F$-factor of an infinite word is a finite word restricted to $F$. Then the pattern complexity over a pattern $\mathcal{P}$ counts the number of distinct $F$-factors of an infinite word, for $P\in \mathcal{P}$. Two finite words are called abelian equivalent if for each letter of the alphabet, they contain the same numbers of occurrences of this letter. The abelian pattern complexity counts the number of $F$-factors up to abelian equivalence. As the main result of the paper, we characterize two-dimensional convex patterns with the following property: if abelian pattern complexity over a pattern $\cal{P}$ is equal to 1, then the word is fully periodic. Similar result holds for a function on $\mathbb{Z}^2$ instead of a word and for constant sums instead of abelian complexity equal to 1. In dimensional 1, we characterize patterns for which there exist non-constant functions with constant sums.