Conjugates of Pisot numbers

Kevin G. Hare, Nikita Sidorov

Published 2020-10-04Version 1

In this paper we investigate the Galois conjugates of a Pisot number $q \in (m, m+1)$, $m \geq 1$. In particular, we conjecture that for $q \in (1,2)$ we have $|q'| \geq \frac{\sqrt{5}-1}{2}$ for all conjugates $q'$ of $q$. Further, for $m \geq 3$, we conjecture that for all Pisot numbers $q \in (m, m+1)$ we have $|q'| \geq \frac{m+1-\sqrt{m^2+2m-3}}{2}$. 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.