Polynomial estimates, exponential curves and Diophantine approximation

2010/09/22 by Dan Coman, Coman, Dan, Evgeny A. Poletsky +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #math.CV #msc:11A55 #msc:11J99 #msc:30D15 #msc:41A17

paper · pdf · doi:10.48550/arxiv.1009.4408

12 pages. To appear in Mathematical Research Letters

arxiv created 2010/09/22 · arxiv updated 2010/09/23

Abstract

Let α∈(0,1)∖\Bbb Q and K=\(e^z,e^αz): |z|≤1\⊂\Bbb C². If P is a polynomial of degree n in \Bbb C², normalized by ‖P‖_K=1, we obtain sharp estimates for ‖P‖_Δ² in terms of n, where Δ² is the closed unit bidisk. For most α, we show that sup_P‖P‖_Δ²≤exp(Cn²log n). However, for α in a subset \mathcal S of the Liouville numbers, sup_P‖P‖_Δ² has bigger order of growth. We give a precise characterization of the set \mathcal S and study its properties.

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