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

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 $\|e^{i\lambda\varphi}\|_{A(\mathbb{T}^2)}\gtrsim |\lambda|$.