Bernstein-Walsh inequalities in higher dimensions over exponential curves

Abstract

Let $\mathbf{x}=(x_1,\dots ,x_d)\in [-1,1{]}^d$ be linearly independent over $\mathbb{Z}$, set $K=\{(e^z,e^{x_{1}z},e^{x_{2}z}\dots ,e^{x_{d}z}):|z|\le 1\}.$ We prove sharp estimates for the growth of a polynomial of degree $n$, in terms of $$E_n(\mathbf{x}):=\sup \{\Vert P{\Vert }_{{\Delta }^{d+1}}:P\in {\mathcal{P}}_n(d+1),\Vert P{\Vert }_K\le 1\},$$ where ${\Delta }^{d+1}$ is the unit polydisk. For all $\mathbf{x}\in [-1,1{]}^d$ with linearly independent entries, we have the lower estimate $$\log E_n(\mathbf{x})\ge \frac{n^{d+1}}{(d-1)!(d+1)}\log n-O(n^{d+1});$$ for Diophantine $\mathbf{x}$, we have $$\log E_n(\mathbf{x})\le \frac{n^{d+1}}{(d-1)!(d+1)}\log n+O(n^{d+1}).$$ In particular, this estimate holds for almost all $\mathbf{x}$ with respect to Lebesgue measure.