Multidimensional cellular automata and generalization of Fekete's lemma

TL;DR

This paper generalizes Fekete's lemma to multidimensional functions and applies it to characterize the information loss in non-surjective cellular automata based on their growth rates.

Contribution

It introduces a multidimensional extension of Fekete's lemma and uses it to analyze the information dynamics of cellular automata.

Findings

Non-surjective cellular automata lose arbitrarily large information on finite supports.

The growth rate of information loss exceeds the boundary growth determined by the automaton's neighborhood.

The new lemma provides a tool for understanding multidimensional combinatorial structures.

Abstract

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.