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Parry order of Parry numbers

Kevin G Hare, Hachem Hichri

Abstract

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.

Parry order of Parry numbers

Abstract

We introduce the \emph{Parry order} , defined as the largest integer for which is a Parry number. This leads to a natural partition of the set of Perron numbers as follows: where is the class of Perron numbers with Parry order , and 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 in terms of algebraic properties of~. We provide explicit examples of non-Parry Perron numbers whose powers become Parry, demonstrating that several 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 -expansion theory with classical questions surrounding Salem numbers and Lehmer-type conjectures.
Paper Structure (9 sections, 16 theorems, 18 equations, 1 figure, 5 tables)

This paper contains 9 sections, 16 theorems, 18 equations, 1 figure, 5 tables.

Table of Contents

  1. A dynamical version of strong Lehmer's conjecture
  2. Dynamical dispersion of Perron numbers
  3. Parry order of a Perron number
  4. New partition of the set $\mathbb{P}$
  5. Computation of Parry order
  6. Some comments on $\beta_d^k$ and $\theta_d^k$
  7. Cubic Perron numbers
  8. Some further data
  9. Conclusion

Key Result

Lemma 1.1

The set of non-simple Parry numbers is dense in $(1,\infty)$.

Figures (1)

  • Figure 2.1: Partition of $\mathbb{P}=\overline{\mathbb{P}}_\Phi\cup\mathbb{P}_\Phi=\mathbb{P}_a\cup \mathcal{P}=\mathcal{H}_\infty \cup (\bigcup_{k\geq 0}{\mathcal{H}_k})$

Theorems & Definitions (46)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Remark 1.3
  • Corollary 1.4
  • proof
  • Remark 1.5
  • Conjecture 1.6
  • Lemma 2.1
  • ...and 36 more