Beta-expansions of \(p\)-adic numbers (Q2805068)

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 \(\mathbb Q(\beta)\cap\mathbb Z_p\). \textit{A. Bertrand} [C. R. Acad. Sci., Paris, Sér. A 285, 419--421 (1977; Zbl 0362.10040)] and \textit{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.