# Polynomial estimates, exponential curves and Diophantine approximation - Mathematics > Complex Variables

Abstract: Let $\alpha\in0,1\setminus{\Bbb Q}$ and $K=\{e^z,e^{\alphaz}:\,|z|\leq1\}\subset{\Bbb C}^2$. If $P$ is a polynomial of degree $n$ in${\Bbb C}^2$, normalized by $\|P\| K=1$, we obtain sharp estimates for$\|P\| {\Delta^2}$ in terms of $n$, where $\Delta^2$ is the closed unit bidisk.For most $\alpha$, we show that $\sup P\|P\| {\Delta^2}\leq\expCn^2\log n$.However, for $\alpha$ in a subset ${\mathcal S}$ of the Liouville numbers,$\sup P\|P\| {\Delta^2}$ has bigger order of growth. We give a precisecharacterization of the set ${\mathcal S}$ and study its properties.

Author: Dan Coman, Evgeny A. Poletsky

Source: https://arxiv.org/