On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

Abstract

Let n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn(X) = −1 + X + Xn. The sequence of Perron numbers $(\theta _n^{ - 1} )_{n \ge 2} $ tends to 1. We prove that the Conjecture of Lehmer is true for $\{ \theta _n^{ - 1} |n \ge 2\} $ by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots θn, zj,n, of Gn(X) lying in |z| < 1, as a function of n, j only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures ${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$ of the trinomials Gn as a function of n only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for $\{ \theta _n^{ - 1} |n \ge 2\} $, with a minoration of the house , and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots of Gn, near the unit circle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.

Alphabets, rewriting trails and periodic representations in algebraic bases

“…In Figure 3 to Figure 6 the Mahler measures M j = M(S * γ j ) of the jth-polynomial sections S * γ j of f β (z) are represented for various θ −1 n < β < θ −1 n−1 , as a function of the number j of monomials added to −1 + x + x n , for different values of n: n = 12, 77, 149, 220. The initial value is M(−1 + x + x n ) ≈ 1.38 by [9], [29]. The growth rates are close to obey a linear growth with j.…”

A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus

“…2 , +2 arcsin κ 2 ]} on the unit circle (Proposition 5.12, Remark 5.13 (i), Theorem 5.15, [VG6] Theorem 6.2).…”

“…In [VG6], the problem of Lehmer for the family (θ −1 n ) n≥2 was solved using the Poincaré asymptotic expansions of the roots of (G n ) of modulus < 1 and of the Mahler measures (M(θ −1 n )) n≥2 . The purpose of the present note is to extend this method to any algebraic integer β of dynamical degree dyg(β) large enough, to show that this method allows to prove that the Conjecture of Lehmer is true in general.…”

“…The minoration (1.0.11) is general and admits a much better lower bound, in a similar formulation, when α only runs over the set of Perron numbers (θ −1 n ) n≥2 . Indeed, in [VG6] , it is shown that m s the Dirichlet L-series for the character χ 3 , with χ 3 the uniquely specified odd character of conductor 3 (χ 3 (m) = 0, 1 or −1 according to whether m ≡ 0, 1 or 2 (mod 3), equivalently χ 3 (m) = m 3 the Jacobi symbol). Numerically the constant Λ r µ r S/(2π) = 0.0315536 .…”

“…In §7 a Theorem of fracturability of the minimal integer polynomials of small Mahler measure is obtained which readily implies the Conjecture of Lehmer for Salem numbers, the Bogomolov property for totally real algebraic numbers. In §3 are presented the "à la Poincaré" technics of asymptotic expansions transposed (and adapted) to Number Theory from Celestial Mechanics; they were introduced for the first time in [VG6] on the height one trinomials G n and G * n , their roots, and their Mahler measures. These asymptotic expansions are fundamental for dealing with the general cases of algebraic integers in §5, §6 and §7.…”