A simple proof for a particular case of Szabados’s conjecture

Resumo

In this work, we present a simple proof for Szabados’s conjecture in the case that η = η1+η2+η3 is a minimal periodic decomposition. The main idea for the proof is the following: let p be a prime large enough so that A ⊂ {0, 1, 2, . . . , p−1} and consider the periodic decomposition η̄ = η̄1+η̄2+η̄3, where the bar denotes the congruence modulo p. Since each configuration η̄i is defined on a finite alphabet, it is easy to see that a line through the origin containing a period for η̄i is nonexpansive. By the pigeonhole principle, we may extend this nonexpansiveness from η̄i to η̄. The result follows from the fact that η̄ and η are essentially the same configurations.