Geodesic
On the recognizability of self-generating sets
Journal of integer sequences, Tome 13 (2010) no. 2
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 $k\ge 2$.
Classification : 68Q45, 68R15, 11B85
Keywords: self-generating set, recognizability, numeration systems, multiplicatively independent integers, Fibonacci word 
@article{JIS_2010__13_2_a4,
     author = {K\"arki,  Tomi and Lacroix,  Anne and Rigo,  Michel},
     title = {On the recognizability of self-generating sets},
     journal = {Journal of integer sequences},
     year = {2010},
     volume = {13},
     number = {2},
     zbl = {1186.68261},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2010__13_2_a4/}
}
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Kärki,  Tomi; Lacroix,  Anne; Rigo,  Michel. On the recognizability of self-generating sets. Journal of integer sequences, Tome 13 (2010) no. 2. http://geodesic.mathdoc.fr/item/JIS_2010__13_2_a4/