Abelian Nivat's conjecture for non-rectangular patterns

Abstract

In this paper, we study the relation between periodicity of two-dimensional words and their abelian pattern complexity. A pattern P in Z n is the set of all translations of some finite subset F of Z n . An F -factor of an infinite word is a finite word restricted to F . Then the pattern complexity over a pattern P counts the number of distinct F -factors of an infinite word, for P ∈ 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 Ffactors 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 P is equal to 1, then the word is fully periodic. Similar result holds for a function on 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.

On forced periodicity of perfect colorings

“…In this section we prove a statement concerning forced periodicity of two-dimensional configurations of low abelian complexity which generalizes a result in [7]. In fact, as in [7] we generalize the definition of abelian complexity from finite patterns to polynomials and prove a statement of forced periodicity under this more general definition of abelian complexity. Let c ∈ {e 1 , .…”

“…In fact, they are corollaries of the theorem. The first part of the theorem was also mentioned in [7] in a slightly different context and in a more general form.…”

“…Note that a configuration of low abelian complexity is not necessarily periodic. Indeed, in [30] it was shown that there exist non-periodic two-dimensional configurations that have abelian complexity A(c, D) = 1 for some finite shape D. However, in [7] it was shown that if A(c, f ) = 1 and if the polynomial f has no line polynomial factors, then c is two-periodic assuming that the support of f is convex. The following theorem strengthens this result and shows that the convexity assumption of the support of the polynomial is not needed.…”

“…Remark. In [7] a polynomial f ∈ Z[X ±1 ] is called abelian rigid if an integral configuration c having low abelian complexity with respect to f implies that c is strongly periodic. In the above theorem we proved that if a polynomial f ∈ Z[x ±1 , y ±1 ] has no line polynomial factors then it is abelian rigid.…”

On perfect coverings of two-dimensional grids

“…Applying Theorems 2 and 5 we have the following theorem that gives sufficient conditions for every (D, b, a)-covering to be periodic for a finite and convex D. The first part of the theorem was also mentioned in [4] in a more general form. Then there exists an algorithm to determine whether there exist any (D, b, a)-coverings.…”