Multidimensional extension of the Morse–Hedlund theorem

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 ≥ 2 . A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of Z d definable by a first order formula in the Presburger arithmetic 〈 Z ; < , + 〉 . 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 Z d definable in 〈 Z ; < , + 〉 in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often.