Quantitative aspects of the Beurling--Helson theorem: Phase functions of a special form

TL;DR

This paper investigates the growth of Fourier algebra norms of phase functions of a specific form on the torus, establishing lower bounds as the phase parameter tends to infinity, which advances understanding of harmonic analysis on compact groups.

Contribution

It provides new lower bounds for the Fourier algebra norms of exponential phase functions with a particular structure, extending the quantitative understanding of the Beurling--Helson theorem.

Findings

Lower bounds for (\u211d^d) norms as ur o 

For (x)|y| phase, -norm grows at least linearly with ur

Results apply to nonconstant real functions a in ()

Abstract

We consider the space $A(\mathbb{T}^d)$ of absolutely convergent Fourier series on the torus $\mathbb{T}^d$. The norm on $A(\mathbb{T}^d)$ is naturally defined by $\|f\|_{A}=\|\widehat{f}\|_{l^1}$, where $\widehat{f}$ is the Fourier transform of a function $f$. For real functions $\varphi$ of a certain special form on $\mathbb T^d, \,d\geq 2,$ we obtain lower bounds for the norms $\|e^{i\lambda\varphi}\|_A$ as $\lambda\rightarrow\infty$. In particular, we show that if $\varphi(x, y)=a(x)|y|$ for $|y|\leq\pi$, where $a\in A(\mathbb{T})$ is an arbitrary nonconstant real function, then $\|e^{i\lambda\varphi}\|_{A(\mathbb{T}^2)}\gtrsim |\lambda|$.