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Communication Dans Un Congrès Année : 2021

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.
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Dates et versions

hal-03168409 , version 1 (23-03-2021)

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  • HAL Id : hal-03168409 , version 1

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Jean-Louis Verger-Gaugry. Recent Progress on the Dynamical Zeta Function of the β -shift, and Lehmer's Problem. 7th International Conference on Uniform Distribution Theory, Friedrich Pillichshammer, JKU Linz, Chair, Peter Kritzer, RICAM Linz, László Mérai, RICAM Linz, Arne Winterhof, RICAM Linz, Feb 2021, Linz, Austria. ⟨hal-03168409⟩
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