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An approximation property of Pisot numbers. (English) Zbl 0962.11034
Given a positive real number $$q$$ and an integer $$m\geq 1$$, denote by $$\Lambda= \Lambda_m$$ the set of all real numbers $$y$$ having at least one representation of the form $y= \varepsilon_0+ \varepsilon_1q+ \varepsilon_2 q^2+\cdots+ \varepsilon_n q^n$ with some integer $$n> 0$$ and $$\varepsilon_i\in \{-m, -m+1,\dots, -1,0,1,\dots, m-1,m\}$$, and let $l^m(q)= \inf\{|y|: y\in \Lambda_m,\;y\neq 0\}.$ It is known that $$q$$ is a Pisot number if and only if $$l^m(q)> 0$$ for all $$m$$. In this paper the authors determine formulae for $$l^m(A)$$ for all $$m$$, where $$A$$ is the golden ratio $$(\sqrt{5}+1)/2$$.

##### MSC:
 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11A67 Other number representations
##### Keywords:
representation; Pisot number; golden ratio
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##### References:
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