Balance properties of multi-dimensional words

A word u is called 1-balanced if for any two factors u and w of u of equal length, we have -1 less than or equal to /v/(i) - /w/(i) less than or equal to 1 for each letter i, where /v/(i) denotes the number of occurrences of i in the factor v. The aim of this paper is to extend the notion of balance to multi-dimensional words. We first characterize all 1-balanced words on Z(n). In particular, we prove they are fully periodic for n > 1. We then give a quantitative measure of non-balancedness for some words on Z(2) with irrational density, including two-dimensional Sturmian words. (C) 2002 Elsevier Science B.V. All rights reserved.