论文标题
PISOT数字的共轭物
Conjugates of Pisot numbers
论文作者
论文摘要
在本文中，我们调查了Pisot Number $ Q \ in（m，m+1）$，$ m \ geq 1 $的Galois coogates。特别是，我们猜想，对于$ q \ in（1,2）$，我们有$ | q'| \ geq \ frac {\ sqrt {5} -1} {2} $ for $ q $ of $ q $。此外，对于$ m \ geq 3 $，我们猜想对于所有PISOT编号$ q \ in（m，m+1）$，我们有$ | q'| \ geq \ frac {m+1- \ sqrt {m^2+2m-3}}} {2} $。如果以$ M = 2 $制作，则类似的猜想。我们猜想所有这些界限都很紧。我们为此猜想提供了部分支持证据。这些证据既是理论和计算本质。 最后，我们将这个猜想与由$β$参数化的Bernoulli卷积的尺寸联系起来，$β$的共轭是PISOT数字的倒数。
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 $β$, whose conjugate is the reciprocal of a Pisot number.
