On the Recognizability of Self-generating Sets
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.