## $$\varepsilon$$-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

### Keywords:

Pisot numbers; Salem numbers; real fields; PV-numbers
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### References:

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