Beta-expansions of \(p\)-adic numbers. (English) Zbl 1343.11066
In this paper, the authors deal with beta-expansions in the ring of the \(p\)-adic integers. They characterize the set of numbers with eventually periodic and finite expansions. In particular, they prove that for \(\beta\) a Pisot-Chabauty number, the set of eventually periodic beta-expansions is \(\mathbb Q(\beta)\cap\mathbb Z_p\). A. Bertrand [C. R. Acad. Sci., Paris, Sér. A 285, 419–421 (1977; Zbl 0362.10040)] and K. Schmidt [Bull. Lond. Math. Soc. 12, 269–278 (1980; Zbl 0494.10040)] proved that if \(\beta\) is Pisot, then the set of eventually periodic beta-expansions consists of the non-negative elements of \(\mathbb Q(\beta)\). Schmidt [loc. cit.] proved a partial converse of this statement. The authors prove an equivalent result to Schmidt’s partial converse. They characterize the set of finite beta-expansions for a family of Pisot-Chabauty numbers.
Reviewer: Takao Komatsu (Wuhan)
MSC:
| 11K16 | Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. |
| 11K41 | Continuous, \(p\)-adic and abstract analogues |
| 37P20 | Dynamical systems over non-Archimedean local ground fields |
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