A Beurling-Carleson set which is a uniqueness set for a given weighted space of analytic functions

Let $p = \big( p(n) \big)_{n \geq 0}$ be a sequence of positive real numbers. We define $B_{p}$ as the space of functions $f$ which are analytic in the unit disc $\mathbb{D}$, continuous on $\overline{\mathbb{D}}$ and such that

$\big\Vert f \big\Vert _{p} := \sum_{n=0}^{+\infty} \vert \hat{f}(n) \vert \, p(n) < +\infty,$

where $\hat{f}(n)$ is the $n^{\textrm{th}}$ Fourier coefficient of the restriction of $f$ to the unit circle $\mathbb{T}$. Let $E$ be a closed subset of $\mathbb{T}$. We say that $E$ is a Beurling-Carleson set if

$\int_{0}^{2\pi} \log^{+} \frac{1}{d(e^{it},E)} \mathrm{d} t < +\infty,$

where $d(e^{it},E)$ denotes the distance between $e^{it}$ and $E$. In 1980, A. Atzmon asked whether there exists a sequence $p$ of positive real numbers such that $\lim_{n \rightarrow +\infty} \frac{p(n)}{n^{k}} = +\infty$ for all $k \geq 0$ and that has the following property: for every Beurling-Carleson set $E$, there exists a non-zero function in $B_{p}$ that vanishes on $E$. In this note, we give a negative answer to this question.