Polynomial estimates, exponential curves and Diophantine approximation

Abstract

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

12 pages. To appear in Mathematical Research Letters

References in corpus (1)

• On Local Behavior of Holomorphic Functions Along Complex Submanifolds of C^N