Bernstein-Walsh inequalities in higher dimensions over exponential curves

TL;DR

This paper establishes sharp growth estimates for polynomials over exponential curves in higher dimensions, revealing precise asymptotic behavior depending on the Diophantine properties of the vector.

Contribution

It provides the first sharp bounds for polynomial growth over exponential curves in multiple dimensions, extending classical Bernstein-Walsh inequalities.

Findings

Lower bound of rac{n^{d+1}}{(d-1)!(d+1)} ext{log} n - O(n^{d+1})

Upper bound of rac{n^{d+1}}{(d-1)!(d+1)} ext{log} n + O(n^{d+1})

Results hold for almost all vectors in the Lebesgue measure sense

Abstract

Let ${{\bf x}}=(x_1,\dots,x_d) \in [-1,1]^d$ 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 $$E_n({\bf x}):=\sup\{\|P\|_{\Delta^{d+1}}:P \in \mathcal P_n(d+1), \|P\|_K \le 1\},$$ where $\Delta^{d+1}$ is the unit polydisk. For all ${{\bf x}} \in [-1,1]^d$ with linearly independent entries, we have the lower estimate $$\log E_n({\bf x})\ge \frac{n^{d+1}}{(d-1)!(d+1)} \log n - O(n^{d+1});$$ for Diophantine $\bf x$, we have $$\log E_n({\bf x})\le \frac{ n^{d+1}}{(d-1)!(d+1)}\log n+O( n^{d+1}).$$ In particular, this estimate holds for almost all $\bf x$ with respect to Lebesgue measure.