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