Parry order of Parry numbers

1. Problem Statement and Motivation

The paper addresses the classification of Perron numbers (real algebraic integers $\beta > 1$ strictly larger in modulus than their other Galois conjugates) based on their dynamical properties related to $\beta$-expansions.

• Context: A number $\beta > 1$ is a Parry number if the orbit of $1$ under the $\beta$-transformation $T_\beta(x) = \beta x \mod 1$ is finite (or eventually periodic). The set of Parry numbers ($\mathcal{P}$) is a subset of Perron numbers ($\mathcal{P}$).
• The Gap: While it is known that all Pisot numbers are Parry numbers and that all Salem numbers are conjectured to be Parry numbers (Schmidt's Conjecture), the algebraic characterization of Parry numbers remains an open problem. Specifically, it is unclear how the "Parry property" behaves under exponentiation.
• Core Question: For a given Perron number $\beta$, how many of its powers $\beta^n$ are Parry numbers? Can a non-Parry Perron number have powers that are Parry numbers?

2. Methodology

The authors employ a combination of algebraic number theory, dynamical systems theory, and computational experimentation.

• Theoretical Framework:
  • Utilization of Solomyak's Criteria: A Perron number with a conjugate of modulus $> \theta_0 = \frac{1+\sqrt{5}}{2}$ (the golden ratio) cannot be a Parry number.
  • Utilization of Boyd's Criteria: Relates the tail coefficients of the companion polynomial to the non-Parry nature of a number.
  • Mahler Measure ($M(\beta)$): Used to bound the growth of conjugates.
  • Galois Theory: Analyzing the action of the Galois group on the conjugates of $\beta^n$ to determine if they exceed the Parry threshold.
• Definitions Introduced:
  • Parry Spectrum ($\text{Spec}_\mathcal{P}(\beta)$): The set of integers $n$ such that $\beta^n$ is a Parry number.
  • Parry Order ($\text{Ord}_\mathcal{P}(\beta)$): The largest integer $n$ such that $\beta^n$ is a Parry number. If $\beta$ is Pisot or Salem, $\text{Ord}_\mathcal{P}(\beta) = \infty$.
• Computational Approach:
  • Explicit calculation of $\beta$-expansions for specific families of polynomials (e.g., $x^d - x - 1$ and $x^d - x^{d-1} - 1$).
  • Verification of Akiyama's test for non-Parry numbers.
  • Generation of large datasets of Perron numbers with bounded degree and coefficient size to analyze density and distribution.

3. Key Contributions and Results

A. Characterization of Infinite Parry Powers

The paper establishes a definitive dynamical characterization of Pisot and Salem numbers:

• Theorem 1.2: A Perron number $\beta$ has infinitely many powers $\beta^n$ that are Parry numbers if and only if $\beta$ is a Pisot or a Salem number.
• Implication: For any Perron number that is not Pisot or Salem, the set of Parry powers is finite.

B. Upper Bounds on Parry Order

For non-Pisot/Salem Perron numbers, the authors derive explicit upper bounds for $\text{Ord}_\mathcal{P}(\beta)$:

• Theorem 2.2: $\text{Ord}_\mathcal{P}(\beta) \leq \frac{d \log(\theta_0)}{\log(M(\beta)) - \log(\beta)}$, where $d$ is the degree and $M(\beta)$ is the Mahler measure.
• Lemma 2.3: A tighter bound based on the largest conjugate $\gamma$ of $\beta$ with $|\gamma| > 1$: $\text{Ord}_\mathcal{P}(\beta) \leq \lceil \frac{\log \theta_0}{\log |\gamma|} \rceil$.
• These bounds prove that for non-Pisot/Salem numbers, $\beta^n$ eventually becomes a non-Parry number for all sufficiently large $n$.

C. Discovery of Non-Parry Numbers with Parry Powers

A major novelty of this work is the construction of the first known examples of non-Parry Perron numbers whose powers are Parry numbers.

• Example 3.1: The authors identify Perron numbers $\beta$ (roots of specific degree 12 polynomials) such that $\beta$ is non-Parry, but $\beta^2$ is Parry.
• This demonstrates that the sets $H_n$ (Perron numbers with Parry order $n$) are non-empty for $n \geq 1$ and that the transition between Parry and non-Parry states is not monotonic.

D. New Partition of Perron Numbers

The authors introduce a new partition of the set of Perron numbers $\mathcal{P}$ based on Parry order:
$$\mathcal{P} = \left( \bigcup_{n \geq 0} H_n \right) \cup H_\infty$$
Where:

• $H_n = \{ \beta \in \mathcal{P} \mid \text{Ord}_\mathcal{P}(\beta) = n \}$
• $H_\infty = S \cup T$ (The set of Pisot and Salem numbers).
• They prove that $H_1$ is a proper subset of Parry numbers and that $H_0$ (non-Parry numbers with no Parry powers) is dense in $(1, \infty)$.

E. Analysis of Specific Families

• Cubic Numbers: The paper provides a detailed analysis of cubic Perron numbers. It proves that for totally real non-Pisot cubic numbers, the Parry order is either 0 or odd.
• Unbounded Order: Theorem 3.13 proves the existence of an infinite family of non-Pisot Parry numbers with unbounded Parry order. Specifically, for the family $x^3 - dx^2 - 2x + d$, the Parry order grows with $d$.
• Polynomial Families:
  • For roots of $x^d - x - 1$ ($d \geq 4$), it is conjectured that all powers are non-Parry.
  • For roots of $x^d - x^{d-1} - 1$, it is conjectured that the root is Parry, but all higher powers are non-Parry.

4. Significance and Implications

1. Bridging Dynamics and Arithmetic: The work connects the combinatorial structure of $\beta$-expansions with the arithmetic geometry of algebraic integers, specifically linking the finiteness of Parry powers to the location of Galois conjugates.
2. Refining Lehmer's Problem: The authors propose a dynamical version of the strong Lehmer's conjecture (Conjecture 1.6): If a Perron number $\beta$ is smaller than the smallest known Salem number ($\tau_{10} \approx 1.176$), then it can have only finitely many Parry powers. This reframes the search for the smallest Salem number as a problem about the distribution of Parry orders.
3. Structural Richness: The discovery that non-Parry numbers can have Parry powers (and vice versa) reveals a complex dynamical landscape previously unobserved. It shows that the property of being a Parry number is not preserved under simple exponentiation for general Perron numbers.
4. Computational Evidence: The paper provides extensive computational data (Table 3.5) showing the density of these classes, suggesting that as the degree increases, the proportion of numbers with finite Parry order ($H_0$) increases, while the proportion of Pisot/Salem numbers ($H_\infty$) decreases.

Conclusion

Hare and Hichri establish that the "Parry property" is a transient dynamical feature for most Perron numbers, persisting only for a finite number of powers unless the number is Pisot or Salem. By defining the Parry Order, they provide a new metric to classify algebraic integers, offering a fresh perspective on classical problems like Schmidt's conjecture and Lehmer's problem. The paper successfully constructs the first examples of non-Parry numbers with Parry powers, proving that the hierarchy of these sets is structurally rich and non-trivial.