Multidimensional cellular automata and generalization of Fekete's lemma

Fekete’s lemma is a well known combinatorial result on number sequences: we extend it to functions defined on $d$-tuples of integers. As an application of the new variant, we show that nonsurjective $d$-dimensional cellular automata are characterized by loss of arbitrarily much information on finite supports, at a growth rate greater than that of the support’s boundary determined by the automaton’s neighbourhood index.