Abstract:

Let $I$ be a finite set of integers and $F$ be a finite set of maps of the form $n\mapsto k_i\, n+\ell_i$ with integer coefficients. For an integer base $k\ge 2$, we study the $k$-recognizability of the minimal set $X$ of integers containing $I$ and satisfying $\varphi(X)\subseteq X$ for all $\varphi\in F$. We answer an open problem of Garth and Gouge by showing that $X$ is $k$-recognizable when the multiplicative constants $k_i$ are all powers of $k$ and additive constants $\ell_i$ are chosen freely. Moreover, solving a conjecture of Allouche, Shallit and Skordev, we prove under some technical conditions that if two of the constants $k_i$ are multiplicatively independent, then $X$ is not $k$-recognizable for any $k\ge 2$.