Totally Real Algebraic Numbers, Bogomolov Property, and Dynamical Zeta Function of the β -shift
Résumé
Schinzel (1973) obtained the lower bound 1/2 Log (1+ sqrt 5/2) = 0.24 . . . for the Weil’s height h(α) of any totally real algebraic integer α \not= 0,
\not= ±1, optimally. This problem of minoration of the height is related to the problem of Lehmer for Salem numbers with Mahler’s
measure M(α). Bombieri and Zannier (2001) introduced the property of Bogomolov for any field F ⊂ Q, by analogy with Bogomolov’s Conjecture: by definition F has the property of Bogomolov relative to h if and only if h(α) = 0 or admits a lower bound > 0 for any α ∈ F. Amoroso and Zannier (2000) proved it for K ab , where K is a number field, Bombieri and Zannier (2001) for totally p-adic fields, Habegger (2011) for Q(E tors ), where E/Q is an elliptic curve. Fili and Miner (2016), using limit equidistribution theorems of Favre and Rivera-Letelier, proved lim inf h(α) ≥ 0.12 . . . for α in the field of totally real algebraic numbers Q tr ; Pottmeyer (2016) obtained the limit infimum lim inf h(α) ≥ 7/(4π)^2 ζ (3) by other techniques.
In this work we show that the dynamical zeta function of the β -shift relative to the arithmetical Rényi-Parry dynamical system, with β = α , α ∈ Q tr , allows to prove that the property of Bogomolov for Q tr is true, with a global explicit minoration.
Domaines
Théorie des nombres [math.NT]
Origine : Fichiers produits par l'(les) auteur(s)