Polynomial estimates, exponential curves and Diophantine approximation

Abstract

Let $α\in(0,1)\setminus{\Bbb Q}$ and $K=\{(e^z,e^{αz}):\,|z|\leq1\}\subset{\Bbb C}^2$. If $P$ is a polynomial of degree $n$ in ${\Bbb C}^2$, normalized by $\|P\|_K=1$, we obtain sharp estimates for $\|P\|_{Δ^2}$ in terms of $n$, where $Δ^2$ is the closed unit bidisk. For most $α$, we show that $\sup_P\|P\|_{Δ^2}\leq\exp(Cn^2\log n)$. However, for $α$ in a subset ${\mathcal S}$ of the Liouville numbers, $\sup_P\|P\|_{Δ^2}$ has bigger order of growth. We give a precise characterization of the set ${\mathcal S}$ and study its properties.