Remez-Type Inequality for Discrete Sets

The classical Remez inequality bounds the maximum of the absolute value of a polynomial $P(x)$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z$ of positive measure in $[-1,1]$. Similarly, in several variables the maximum of the absolute value of a polynomial $P(x)$ of degree $d$ on the unit cube $Q^n_1 \subset {\mathbb R}^n$ can be bounded through the maximum of its absolute value on any subset $Z\subset Q^n_1$ of positive $n$-measure. The main result of this paper is that the $n$-measure in the Remez inequality can be replaced by a certain geometric invariant $\omega_d(Z)$ which can be effectively estimated in terms of the metric entropy of $Z$ and which may be nonzero for discrete and even finite sets $Z$.