Conjugates of Pisot numbers

TL;DR

This paper investigates the properties of Galois conjugates of Pisot numbers, proposing conjectures on their bounds, providing partial evidence, and connecting these to Bernoulli convolution dimensions.

Contribution

It formulates new conjectures on the bounds of conjugates of Pisot numbers and offers partial theoretical and computational support for these conjectures.

Findings

Proposes bounds for conjugates of Pisot numbers in various intervals.

Provides partial theoretical and computational evidence supporting the conjectures.

Connects the conjectures to the dimension of Bernoulli convolutions.

Abstract

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.