Recent Progress on the Dynamical Zeta Function of the β -shift, and Lehmer's Problem
Résumé
A class of lacunary integer polynomials, called almost Newman polynomials, re-lated to Newman polynomials, is studied. We show that this class has strikingproperties, and allows the investigation of the problem of Lehmer by the dynamicalzeta function of the beta-shift. The question of the localization of its lenticular polesinside Solomyak’s fractal, viewed as a neighbourhood of the unit circle in the com-plex plane, is conducted in this framework. This class appears as a pertinent toolto investigate the part of the roots of reciprocal integer polynomials of small Mahlermeasure, in the cusp; the importance of the cusp had been guessed by Langevin.This study is based on the role played by periodic representations in algebraic basis of real number fields (Baker, Sidorov, Kala, V ́avra, Pethoe, Frougny, Mas ́akov ́a,Pelantov ́a, Svobodov ́a, Kov ́acs, K ̈ornyei), with variable bases. The problem of limitMahler measures (Deninger) is evoked in this context.
Origine : Fichiers produits par l'(les) auteur(s)