[Paper Review] A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus

Jean-Louis Verger-Gaugry|arXiv (Cornell University)|Sep 12, 2017

Mathematical Dynamics and Fractals313 references3 citations

TL;DR

This paper proves Lehmer's and Schinzel-Zassenhaus's conjectures on the lower bounds of Mahler measures of non-cyclotomic algebraic integers using dynamical zeta functions, Parry upper functions, and Poincaré asymptotic expansions of poles. It establishes a continuous lenticular minorant for Mahler measure and shows that the infimum of Mahler measures is bounded below by the Perron number $\theta^{-1}_{31} \approx 1.08545$, the dominant root of $z^{31} - z^{30} - 1$, and proves equidistribution of Galois orbits toward the unit circle as dynamical degree increases.

ABSTRACT

The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{house(α)}(z)$ associated with the dynamical zeta functions $ζ_{house(α)}(z)$ of the Rényi--Parry arithmetical dynamical systems, for $α$ an algebraic integer $α$ of house "$house(α)$" greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{house(α)}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $house(α)$ tends to $1^+$, giving rise to a continuous lenticular minorant ${ m M}_{r}(house(α))$ of the Mahler measure ${ m M}(α)$, (iii) the Poincaré asymptotic expansions of these poles and of this minorant ${ m M}_{r}(house(α))$ as a function of the dynamical degree. With the same arguments the conjecture of Schinzel-Zassenhaus is proved to be true. An inequality improving those of Dobrowolski and Voutier ones is obtained. The set of Salem numbers is shown to be bounded from below by the Perron number $θ_{31}^{-1} = 1.08545\ldots$, dominant root of the trinomial $-1 - z^{30} + z^{31}$. Whether Lehmer's number is the smallest Salem number remains open. A lower bound for the Weil height of nonzero totally real algebraic numbers, $ eq \pm 1$, is obtained (Bogomolov property). For sequences of algebraic integers of Mahler measure smaller than the smallest Pisot number, whose houses have a dynamical degree tending to infinity, the Galois orbit measures of conjugates are proved to converge towards the Haar measure on $|z|=1$ (limit equidistribution).

Motivation & Objective

• To resolve Lehmer's conjecture by proving the existence of a universal positive lower bound for the Mahler measure of non-cyclotomic algebraic integers.
• To establish the Schinzel-Zassenhaus conjecture by deriving a uniform lower bound for the Weil height of totally real algebraic numbers not equal to $\pm1$.
• To demonstrate that the set of Salem numbers is bounded below by the Perron number $\theta^{-1}_{31} \approx 1.08545$, the dominant root of $z^{31} - z^{30} - 1$.
• To show that Galois orbit measures of conjugates of algebraic integers with Mahler measure below the smallest Pisot number equidistribute toward the Haar measure on $|z| = 1$ as the dynamical degree tends to infinity.

Proposed method

• Introduces the dynamical degree $\text{dyg}(\beta)$ as an integer function of $\beta > 1$, defined via intervals bounded by roots $\theta^{-1}_n$ of trinomials $G_n(z) = -1 + z + z^n$, which increases as $\beta \to 1^+$.
• Analyzes the poles of the Rényi-Parry dynamical zeta functions $\zeta_\alpha(z)$, identifying lenticuli of poles that equidistribute on a limit arc of the unit circle as $\alpha \to 1^+$.
• Applies Poincaré asymptotic expansions to the poles and to the minorant $M_r(\alpha)$ of the Mahler measure $M(\alpha)$, expressing them as functions of the dynamical degree $\text{dyg}(\alpha)$.
• Uses the Parry Upper function $f_\alpha(z)$ associated with $\zeta_\alpha(z)$ to derive a continuous minorant $M_r(\alpha)$ that uniformly bounds $M(\alpha)$ from below.
• Applies transition region analysis to the roots $z_{j,n}$ of $G_n(z)$, dividing the complex plane into angular sectors based on $\text{arg}(z) \asymp \frac{2\pi \log n}{n}$ and $\text{arg}(z) \asymp \frac{2\pi \sqrt{\log n \log \log n}}{n}$, with precise asymptotic control via sequences $u_n, v_n$.
• Extends results from real $\beta > 1$ to complex $\alpha$ with $|\alpha| > 1$ via the dynamical degree $\text{dyg}(\alpha) := \text{dyg}(|\alpha|)$, reducing the complex case to the real one.

Experimental results

Research questions

• RQ1 Is there a universal positive lower bound for the Mahler measure of any non-cyclotomic algebraic integer?
• RQ2 Does the set of Salem numbers have a positive lower bound, and if so, what is its infimum?
• RQ3 Can the Galois orbit measures of conjugates of algebraic integers with small Mahler measure equidistribute on the unit circle as the dynamical degree tends to infinity?
• RQ4 What is the precise asymptotic behavior of the poles of the Rényi-Parry zeta function $\zeta_\alpha(z)$ as $\alpha \to 1^+$?
• RQ5 How do the roots of the trinomials $G_n(z) = -1 + z + z^n$ distribute angularly in the complex plane as $n \to \infty$?

Key findings

• The conjecture of Lehmer is proven true, with a universal lower bound for Mahler measure $M(\alpha) \geq 1 + c$ for all non-cyclotomic algebraic integers $\alpha$, where $c > 0$ is effectively computable.
• The conjecture of Schinzel-Zassenhaus is proven true, yielding a new lower bound for the Weil height of nonzero totally real algebraic numbers not equal to $\pm1$, improving upon Dobrowolski and Voutier's bounds.
• The smallest known Salem number is bounded below by $\theta^{-1}_{31} \approx 1.08545$, the dominant root of the trinomial $z^{31} - z^{30} - 1$, and this value is conjectured to be the infimum of the set of Salem numbers.
• For sequences of algebraic integers with Mahler measure less than the smallest Pisot number and dynamical degree tending to infinity, the Galois orbit measures of their conjugates converge weakly to the normalized Haar measure on the unit circle $|z| = 1$.
• The minorant $M_r(\alpha)$ of the Mahler measure $M(\alpha)$ is shown to be continuous and lenticular, arising from the equidistribution of poles of $\zeta_\alpha(z)$ on a limit arc of the unit circle as $\alpha \to 1^+$.
• The asymptotic expansion of the dynamical degree $\text{dyg}(\beta)$ for $\beta > 1$ near 1 is derived as $\text{dyg}(\beta) = -\frac{\log(\beta - 1)}{\beta - 1} \left(1 + O\left(\left(\frac{\log(-\log(\beta - 1))}{\log(\beta - 1)}\right)^2\right)\right)$, with explicit degree bounds for $\beta$ in the interval $[\theta^{-1}_n, \theta^{-1}_{n-1})$.