## 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
Full Text:

### References:

 [1] A. Rényi, Representations for real numbers and their ergodic properties,Acta Math. Acad. Sci. Hung.,8 (1957), 477–493. · Zbl 0079.08901 [2] W. Parry, On the {$$\beta$$} expansions of real numbers,Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416. · Zbl 0099.28103 [3] A. Bertrand-Mathis, Développements en base {$$\theta$$} et répartition modulo 1,Bulletin de la S. M. F. (1986). [4] A. Bertrand-Mathis, Points génériques pour la mesure de Champernowne sur certains systèmes codés. A pariatre auJournal de Théorie ergodique. [5] A. Bertrand-Mathis, Le {$$\theta$$}-shift sans peine. En préparation. [6] Ito-Shiokawa, A construction of {$$\beta$$}-normal sequence,J. Math. Soc. Japan,27 (1975). · Zbl 0292.10040 [7] Ito-Takahasni, Markov subshift and realisation of {$$\beta$$}-expansions.J. Math. Soc. Japan,26 (1974). [8] F. Hofbauer, {$$\beta$$}-shift have unique maximal measure,Monatshefte für Mathematik,85 (1978), 189–198. · Zbl 0384.28009 [9] D. Champernowne, The construction of decimal normal in the scale of ten,J. of London Math. Soc.,8 (1950), 254–260. · Zbl 0007.33701 [10] A. S. Fraenkel, Systems of numeration,Amer. Math. Monthly,92 (1985), 105–114. · Zbl 0568.10005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.