## How to write integers in a non-integral basis. (Comment écrire les nombres entiers dans une base qui n’est pas entière.)(French)Zbl 0695.10005

Let $$d=(d_ n)$$, $$n\geq 0$$, be a strictly increasing sequence of natural numbers such that $$d_ 0=1$$. Then every natural number $$N$$ can be written in a unique manner as $$N=\sum^{n}_{j=0}m_ jd_ j$$. Then $$L(d)$$ denotes the set of all admissible blocks $$m_ n...m_ 1m_ 0$$ obtained as expansions of natural numbers. It is shown that $$L(d)$$ is the set of all admissible blocks of digits with respect to a $$\beta$$-expansion if and only if $$m_ n...m_ 1m_ 0\in L(d)$$ implies $$m_ n...m_ 1m_ 00\in L(d)$$. In this case $$\beta =\lim_{n\to \infty} d_{n+1}d_ n^{-1}.$$
Reviewer: F. Schweiger

### MSC:

 11A63 Radix representation; digital problems 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 54H20 Topological dynamics (MSC2010)

### Keywords:

digital problem; topological dynamics; $$\beta$$-expansion
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### References:

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