Polynomial estimates, exponential curves and Diophantine approximation

TL;DR

This paper establishes sharp polynomial estimates on exponential curves in complex space, revealing different growth behaviors depending on whether the parameter is a typical real number or a Liouville number.

Contribution

It provides the first sharp bounds for polynomial norms on exponential curves and characterizes the set of parameters with exceptional growth rates.

Findings

For most b5, the polynomial norm grows at most exponentially with n^2 log n.

For Liouville numbers, the growth of polynomial norms is significantly larger.

A precise characterization of the set of Liouville numbers affecting growth rates.

Abstract

Let $\alpha\in(0,1)\setminus{\Bbb Q}$ and $K=\{(e^z,e^{\alpha z}):\,|z|\leq1\}\subset{\Bbb C}^2$. 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 $\sup_P\|P\|_{\Delta^2}\leq\exp(Cn^2\log n)$. 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 precise characterization of the set ${\mathcal S}$ and study its properties.