Cyclicity in weighted ${\mathrm{\ell }}^p$ spaces

We study the cyclicity in weighted ${\mathrm{\ell }}^p(\mathbb{Z})$ spaces. For $p \geq 1$and$\beta \geq 0$,let$\ell^p_\beta($\mathbb{Z}$)$ be the space of sequences $u=(u\mathrm{\_}n)\mathrm{\_}n\in \mathbb{Z}$ such that $(u_n |n|^{\beta})\in \ell^p($\mathbb{Z}$) $. We obtain both necessary conditions and sufficient conditions for $u$ to be cyclic in ${\mathrm{\ell }}^p\mathrm{\_}\beta (\mathbb{Z})$, in other words, for $\{(u\mathrm{\_}n+k)\mathrm{\_}n\in \mathbb{Z},k\in \mathbb{Z}\}$ to span a dense subspace of ${\mathrm{\ell }}^p\mathrm{\_}\beta (\mathbb{Z})$. The conditions are given in terms of the Hausdorff dimension and the capacity of the zero set of the Fourier transform of $u$.