On an extremal problem for polynomials in a multi-dimensional case (Q1276056)

Let \(d\in{\mathbb N}\) be a positive integer, \( {\mathbb T}^d = [-\pi,\pi]^d\) the \(d\)-dimensional torus, \(S\) the central symmetric body from \({\mathbb R}^d\) and \[ T_S({\mathbf x}) = \sum\limits_{k\in S\cap{\mathbb Z}^d} c_ke^{i\mathbf{kx}}\tag{1} \] be trigonometric polynomials with spectrum \(S\). The set of polynomials (1) is denoted by \({\mathcal T}(S)\) and by \({\mathcal T}^0(S)\) the set of polynomials with zero mean. The author denotes \(m(S,d) = \inf\limits_{t\in{\mathcal T}^0}\text{mes}\{{\mathbf x}\in{\mathbb T}^d\mid t({\mathbf x})\geq 0\}\) and \(S_n = \{{\mathbf x}\in{\mathbb R}^d\mid (\frac{x_1}{n_1},\dots,\frac{x_d}{n_d})\in S\}\). The basic result of the paper is the bilateral estimate for \(m(S_n,d)\).