A dimension-free Remez-type inequality on the polytorus

A dimension-free Remez-type inequality on the polytorus, Discrete Analysis 2025:4, 21 pp.

The classical Remez inequality is a powerful tool in approximation theory. It allows one to bound the maximum value of a real polynomial \(p\) on a closed interval \(\Delta\) in \(\mathbb R\) in terms of its supremum on an arbitrary subset \(J\) of \(\Delta\) of positive measure. More precisely, if \(m\) stands for Lebesgue measure, the inequality states that the maximum of \(p\) on \(\Delta\) is at most \((4m(\Delta)/m(J))^d\sup_J|p|\), where \(d\) is the degree of \(p\).

Recently, motivated by important applications to various problems in modern analysis, there has been considerable interest in generalizing the classical Remez inequality to a multivariate setting, that is, replacing the interval by certain compact sets in higher dimensions. The existing literature in this direction mainly deals with convex sets, and the resulting multivariate generalizations of the Remez inequality typically involve constants that depend on dimension.

In general this dependence is necessary, but that does not preclude the possibility that for some interesting classes of sets and subsets one might obtain dimension-independent constants. The main result of this paper is a dimension-free Remez-type inequality for analytic polynomials on the polytorus \(\mathbb T^n\). The subsets in question are of the form \(\Omega_K^n\), where \(\Omega_K\) is the cyclic group of \(K\)th roots of unity. Let \(\alpha\) be a non-negative sequence in \(\mathbb Z^n\). We write \(|\alpha|\) for \(\sum_{i=1}^n\alpha_i\), and if \(z\in\mathbb C^n\), we write \(z^\alpha\) for \(\prod_{i=1}^nz_i^{\alpha_i}\). We also write \([K]\) for \(\{0,1,\dots,K-1\}\). Then an analytic polynomial of degree at most \(d\) and individual degree at most \(K-1\) (where \(K\geq 2\)) means a function \(f\) of the form \(\sum_{\alpha\in[K]^n,|\alpha|\leq d}c_\alpha z^{\alpha}\). The dimension-free Remez inequality proved in this paper states that if \(f\) is an analytic polynomial of degree \(d\) and individual degree at most \(K-1\), then \(\|f\|_{\mathbb T^n}\leq C(d, K)\|f\|_{\Omega_K^n}\), where we write \(\|f\|_X\) to denote the supremum norm of the restriction of \(f\) to \(X\). The key point of interest here is that the constant \(C(d,K)\) depends only on \(d\) and \(K\), and not on \(n\).

This is a noteworthy new result obtained by means of quite complex and original arguments, combining several subtle analytical ingredients.

The authors use this result to provide a new proof of the Bohnenblust-Hille inequality for functions on products of cyclic groups. The original form of the Bohnenblust-Hille inequality states that if \(f\) is a polynomial of degree at most \(d\) defined on \(\mathbb T^n\), then \(\|\hat f\|_{\frac{2d}{d+1}}\leq C(d)\|f\|_\infty\). Note that the Plancherel identity immediately gives the inequality \(\|\hat f\|_2=\|f\|_2\leq\|f\|_\infty\), and since the \(\ell_p\) norms on \(\mathbb Z^n\) increase as \(p\) decreases, the Bohnenblust-Hille inequality is stating that for degree-\(d\) polynomials one can obtain a significant improvement on the trivial bound. A similar inequality was proved by the authors when \(f\) is defined on the group \(\Omega_K^n\), where now the constant depends on \(K\) as well. The main theorem of this paper yields a different proof of this result.

The results of this paper have subsequently been improved by the authors, together with Lars Becker and Ohad Klein. However, the proof here is quite different, and interesting in its own right.