arXiv 2603.195542026-03-20math.NT

帕里数的帕里阶数

Parry order of Parry numbers

Kevin G Hare, Hachem Hichri

AI总结：

本文引入帕里阶数，研究帕里数的性质，并证明帕里数的帕里阶数与Pisot和Salem数相关，同时给出帕里阶数的上界和实例。

摘要：

我们引入帕里阶数Ord_P(β)，定义为使得β^n为帕里数的最大整数n。这导致了Perron数的自然划分：P = (∪_{n≥0} H_n) ∪ H_∞，其中H_n是具有帕里阶数n的Perron数集合，H_∞ = S ∪ T恰好包含所有Pisot和Salem数。我们证明Perron数有无限多个帕里幂当且仅当它是Pisot或Salem数。对于其他Perron数，仅有限个幂可能是帕里数。我们给出了Ord_P(β)的显式上界，基于β的代数性质。我们提供了非帕里Perron数的实例，其幂成为帕里数，证明了多个H_n非空且结构丰富。我们还给出了一个无限族的非Pisot三次数，它们具有有限的帕里阶数，但该族的帕里阶数无界。这些结果为Perron数建立了新的动力学视角，连接β展开理论与经典Salem数和Lehmer型猜想问题。

英文摘要：

We introduce the \emph{Parry order} $\mathrm{Ord}_P(β)$, defined as the largest integer $n$ for which $β^n$ is a Parry number. This leads to a natural partition of the set of Perron numbers as follows: \[ \mathcal{P} = \left( \bigcup_{n \geq 0} H_n \right) \cup H_\infty, \] where $H_n$ is the class of Perron numbers with Parry order $n$, and $H_\infty = S \cup T$ consists exactly of all Pisot and Salem numbers. We show that a Perron number has infinitely many Parry powers if and only if it is Pisot or Salem. For every other Perron number, only finitely many powers can be Parry. We give an explicit upper bound on $\mathrm{Ord}_P(β)$ in terms of algebraic properties of~$β$. We provide explicit examples of non-Parry Perron numbers whose powers become Parry, demonstrating that several $H_n$ are non-empty and structurally rich. We give an infinite family of cubic non-Pisot numbers, all of which have finite Parry order, but where the family has unbounded Parry order. These results establish a new dynamical perspective on Perron numbers, connecting $β$-expansion theory with classical questions surrounding Salem numbers and Lehmer-type conjectures.

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