In this paper, we investigate the Galois conjugates of a Pisot number q ∈ (m,m + 1), m ≥ 1. In particular, we conjecture that for q ∈ (1, 2) we have |q′|≥ 5−1 2 for all conjugates q′ of q. Further, for m ≥ 3, we conjecture that for all Pisot numbers q ∈ (m,m + 1) we have |q′|≥ m+1−m2 +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.

中文翻译：

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皮索数的共轭

在本文中，我们研究了皮索数的伽罗瓦共轭q ∈ (米,米 + 1),米 ≥ 1. 特别是，我们推测对于q ∈ (1, 2)我们有|q'|≥ 5-1 2对于所有共轭q'的q. 此外，对于米 ≥ 3, 我们推测对于所有的皮索数q ∈ (米,米 + 1)我们有|q'|≥ 米+1-米2 +2米-3 2. 类似的猜想米 = 2. 我们推测所有这些界限都是紧密的。我们为这个猜想提供了部分支持证据。该证据具有理论和计算性质。最后，我们将这个猜想与伯努利卷积参数化的维度的结果联系起来β，其共轭是皮索数的倒数。