Let α ε (0,1)\ ℚ and K= ((ez, e αz: z ≤) ⊂ ℂ2 If P is a polynomial of degree n in 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 supP ∥P∥Δ2 <exp(Cn2 log n). However, for in a subset S of the Liouville numbers, supP ∥P∥Δ2 has bigger order of growth. We giveaprecise characterization of the set S and study its properties.
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