Conjugates of Pisot numbers

Abstract

In this paper we investigate the Galois conjugates of a Pisot number $q\in (m,m+1)$, $m\ge 1$. In particular, we conjecture that for $q\in (1,2)$ we have $|q^{\prime }|\ge \frac{\sqrt{5}-1}{2}$ for all conjugates $q^{\prime }$ of $q$. Further, for $m\ge 3$, we conjecture that for all Pisot numbers $q\in (m,m+1)$ we have $|q^{\prime }|\ge \frac{m+1-\sqrt{m^2+2m-3}}{2}$. A similar conjecture if made for $m=2$. We conjecture that all of these bounds are tight. We provide partial supporting evidence for this conjecture. This evidence is both of a theoretical and computational nature. Lastly, we connect this conjecture to a result on the dimension of Bernoulli convolutions parameterized by $\beta$, whose conjugate is the reciprocal of a Pisot number.

17 pages, 2 figures

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