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On a property of Pisot numbers and related questions. (English) Zbl 0923.11148
Summary: Denote by \(q\) a real number satisfying \(1<q<2\). Let \(k\) be a positive integer and let \(0=:y^{(k)}_0<y_1^{(k)}<y_2^{(k)}<\dots\) be the increasing sequence of those real numbers \(y\) which have at least one representation of the form \(y=\varepsilon_0+\varepsilon_1 q+\varepsilon_2 q^2+\dots +\varepsilon_i q^i\), where \(i\geqq 1\) and \(\varepsilon_0,\dots,\varepsilon_i\in\{0,1,\dots,k\}\). We define the difference sequence \((u_n^{(k)})_{n\geqq 0}\) by \(u^{(k)}_n =y^{(k)}_{n+1}-y^{(k)}_n\).
A typical result of the paper is: Theorem 1. The real \(q\in ]1,2[\) is a Pisot number if and only if we have \(\liminf(u^{(k)}_n)>0\) for all positive integers \(k\).

MSC:
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Keywords:
Pisot numbers
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