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 $\mathbb {Z}$, and 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

$$\begin{aligned} E_n(\mathbf{x}):=\sup \{\Vert P\Vert _{\Delta ^{d+1}}:P \in \mathcal {P}_n(d+1), \Vert P\Vert _K \le 1\}, \end{aligned}$$

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

$$\begin{aligned} \log E_n(\mathbf{x})\ge \frac{n^{d+1}}{(d-1)!(d+1)} \log n - O(n^{d+1}); \end{aligned}$$

for Diophantine $\mathbf{x}$, we have

$$\begin{aligned} \log E_n(\mathbf{x})\le \frac{ n^{d+1}}{(d-1)!(d+1)}\log n+O( n^{d+1}). \end{aligned}$$

In particular, this estimate holds for almost all $\mathbf{x}$ with respect to Lebesgue measure. The results here generalize those of Coman and Poletsky [6] for $d=1$ without relying on estimates for best approximants of rational numbers which do not hold in the vector-valued setting.