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 \ZZ^d definable by a first order formula
in the Presburger arithmetic . 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 in terms of some functions counting recurrent
blocks, that is, blocks occurring infinitely often
