The upper density of an automatic set is rational

Bell, Jason P.

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 585-604.

Soient $k\ge 2$ un nombre naturel et $S$ un ensemble $k$-reconnaissable de nombres naturels. On montre que la densité inférieure et la densité supérieure de $S$ sont des nombres rationnels qui sont récursivement calculables, et on donne un algorithme pour les calculer. En outre, on montre que, pour $k\ge 2$ et $(\alpha ,\beta )\in {\mathbb{Q}}^2$ tels que $0<\alpha <\beta <1$ ou $(\alpha ,\beta )\in \left\{\right(0,0),(1,1\left)\right\}$, il existe un sous-ensemble $k$-reconnaissable de $\mathbb{N}$ tel que la densité inférieure et la densité supérieure sont respectivement égales à $\alpha$ et $\beta$. De plus, on montre que ces valeurs sont précisément celles qui peuvent apparaître comme densités inférieure et supérieure d’un ensemble reconnaissable.

Given a natural number $k\ge 2$ and a $k$-automatic set $S$ of natural numbers, we show that the lower density and upper density of $S$ are recursively computable rational numbers and we provide an algorithm for computing these quantities. In addition, we show that for every natural number $k\ge 2$ and every pair of rational numbers $(\alpha ,\beta )$ with $0<\alpha <\beta <1$ or with $(\alpha ,\beta )\in \left\{\right(0,0),(1,1\left)\right\}$ there is a $k$-automatic subset of the natural numbers whose lower density and upper density are $\alpha$ and $\beta$ respectively, and we show that these are precisely the values that can occur as the lower and upper densities of an automatic set.

Classification : 11B85, 68Q45
Mots-clés : upper density, automatic sets, Cobham’s theorem, formal languages

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