On the recognizability of self-generating sets

Abstract

Let I be a finite set of integers and F be a finite set of maps of the form n ↔k i n+ i with integer coefficients. For an integer base k ≥2, we study the k-recognizability of the minimal set X of integers containing I and satisfying Φ (X)⊆X for all Φ ∈ F. In particular, solving a conjecture of Allouche, Shallit and Skordev, we show under some technical conditions that if two of the constants k i are multiplicatively independent, then X is not k-recognizable for any k≤ 2. © 2009 Springer Berlin Heidelberg.