Multidimensional extension of the Morse--Hedlund theorem

TL;DR

This paper extends the classical Morse--Hedlund theorem from one-dimensional sequences to higher dimensions, characterizing multidimensional sets definable in Presburger arithmetic through recurrence properties.

Contribution

It provides a complete multidimensional extension of the Morse--Hedlund theorem using Presburger definability and recurrence functions.

Findings

Characterization of multidimensional periodicity via recurrence functions

Extension of Morse--Hedlund theorem to arbitrary dimensions

Use of Muchnik's criterion for definability in Presburger arithmetic

Abstract

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let $d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of $\ZZ^d$ definable by a first order formula in the Presburger arithmetic $<\ZZ;<,+>$. With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of $\ZZ^d$ definable in $<\ZZ;<,+>$ in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often.