## 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

Pisot numbers
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### References:

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