Bernstein-Walsh inequalities in higher dimensions over exponential curves

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 $E_n({\bf x}):=\sup\{\|P\|_{Δ^{d+1}}:P \in \mathcal P_n(d+1), \|P\|_K \le 1\},$ where $Δ^{d+1}$ 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.

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