A dimension-free discrete Remez-type inequality on the polytorus

TL;DR

This work proves a dimension-free discretization inequality for analytic polynomials on the polytorus: for degree $\le d$ and individual degree $\le K-1$, the sup norm on $\mathbf{T}^n$ is controlled by the sup norm on the finite grid $\Omega_K^n$ with a constant depending only on $d$ and $K$. The authors develop a novel class of Fourier multipliers that act boundedly on low-degree polynomials independent of dimension, and implement a two-step strategy: (1) bound $\|f\|_{\mathbf{T}^n}$ by $\|f\|_{\Omega_{2K}^n}$ via probabilistic lifting and dilation, and (2) descend from $\Omega_{2K}^n$ to $\Omega_K^n$ by analyzing the value at $\sqrt{\omega}$ and decomposing $f$ into inseparable parts with controlled norms. As a consequence, they obtain a dimension-free Bohnenblust--Hille-type inequality for functions on products of cyclic groups and provide a detailed analysis in the prime-$K$ case, where inseparability has a crisp structure. The results contribute a new set of tools (Fourier multipliers, inseparable-part analysis) for discrete harmonic analysis and discrete sampling, with potential implications for related inequalities and discrete quantum settings.

Abstract

Consider $f:Ω^n_K \to \mathbf{C}$ a function from the $n$-fold product of multiplicative cyclic groups of order $K$. Any such $f$ may be extended via its Fourier expansion to an analytic polynomial on the polytorus $\mathbf{T}^n$, and the set of such polynomials coincides with the set of all analytic polynomials on $\mathbf{T}^n$ of individual degree at most $K-1$. In this setting it is natural to ask how the supremum norms of $f$ over $\mathbf{T}^n$ and over $Ω_K^n$ compare. We prove the following \emph{discretization of the uniform norm} for low-degree polynomials: if $f$ has degree at most $d$ as an analytic polynomial, then $\|f\|_{\mathbf{T}^n}\leq C(d,K)\|f\|_{Ω_K^n}$ with $C(d,K)$ independent of dimension $n$. As a consequence we also obtain a new proof of the Bohnenblust--Hille inequality for functions on products of cyclic groups. Key to our argument is a special class of Fourier multipliers on $Ω_K^n$ which are $L^\infty\to L^\infty$ bounded independent of dimension when restricted to low-degree polynomials. This class includes projections onto the $k$-homogeneous parts of low-degree polynomials as well as projections of much finer granularity.

Theorems & Definitions (23)

• Theorem 1
• Remark 1
• Theorem 2
• Remark 2
• Corollary 3: Cyclic-group Bohnenblust--Hille
• Proposition 1: Torus bounded by $\Omega_{2K}$
• Lemma 4
• proof
• proof : Proof of Proposition \ref{['prop:Om2K-boundedness']}
• Proposition 2