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

Abstract

AbstractLetn ≥2 be an integer and denote byθnthe real root in (0, 1) of the trinomialGn(X) = −1 +X+Xn. The sequence of Perron numbers(θn−1)n≥2$(\theta _n^{ - 1} )_{n \ge 2} $tends to 1. We prove that the Conjecture of Lehmer is true for{θn−1|n≥2}$\{ \theta _n^{ - 1} |n \ge 2\} $by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the rootsθn,zj,n, ofGn(X) lying in|z|< 1, as a function ofn, jonly. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measuresM(Gn)=M(θn)=M(θn−1)${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$of the trinomialsGnas a function ofnonly, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for{θn−1|n≥2}$\{ \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 ofGn, near the unit circle, and limit equidistribution of the conjugates, forntending 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.