The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive

TL;DR

This paper proves the Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive, leading to sharper asymptotic estimates for the Bohr radius and Sidon constants in complex analysis.

Contribution

It establishes the hypercontractivity of the Bohnenblust--Hille inequality, improving bounds and deriving new asymptotic behaviors for related constants.

Findings

The Bohnenblust--Hille inequality constant can be taken as C^m for some C>1.

The Bohr radius for the polydisc behaves asymptotically as sqrt((log n)/n).

The Sidon constant for certain frequency sets has a precise asymptotic expression.

Abstract

The Bohnenblust--Hille inequality says that the $\ell^{\frac{2m}{m+1}}$-norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\C^n$ is bounded by $\| P\|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ denotes the supremum norm on the polydisc $\D^n$. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be $C^m$ for some $C>1$. Combining this improved version of the Bohnenblust--Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc $\D^n$ behaves asymptotically as $\sqrt{(\log n)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies $\bigl\{\log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$ as $N\to \infty$.