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\begin{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 $k\ge 2$.
\end{abstract}

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