Bernstein-Walsh inequalities in higher dimensions over exponential curves

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.