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\(\varepsilon\)-Pisot numbers in any real algebraic number field are relatively dense. (English) Zbl 1043.11074

The authors prove the following theorem. Let \(K\) be a real algebraic number field, say of degree \(d\) over the field of rational numbers. Then for any \(\varepsilon>0\) there is a number \(L=L(\varepsilon)\) such that any subinterval of \((1,\infty)\) of length \(L\) contains a Pisot number \(\alpha\) which lies in \(K\), has degree \(d\) and all of whose conjugates except for \(\alpha\) lie in the disc \(| z| < \varepsilon\). The classical version of this result corresponding to \(\varepsilon=1\) is well known. They also derive several interesting corollaries. One of them concerns Salem numbers. They prove that if \(\tau\) is a Salem number then for any any \(\varepsilon>0\) there is a number \(L=L(\varepsilon)\) such that any subinterval of \((1,\infty)\) of length \(L\) contains a real number \(\lambda\) satisfying \(\{\lambda \tau^n\} \in [0,\varepsilon) \cup (1-\varepsilon, 1)\) for each \(n=1,2,\dots\).

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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