Cyclicity in ${\mathrm{\ell }}^p$ spaces and zero sets of the Fourier transforms

We study the cyclicity of vectors $u$ in ${\mathrm{\ell }}^p(\mathbb{Z})$. It is known that a vector $u$ is cyclic in ${\mathrm{\ell }}^2(\mathbb{Z})$ if and only if the zero set, $\mathcal{Z}(\widehat{u})$, of its Fourier transform, $\widehat{u}$, has Lebesgue measure zero and $\log \mathrm{\mid }\widehat{u}\mathrm{\mid }\in ̸L^1(\mathbb{T})$, where $\mathbb{T}$ is the unit circle. Here we show that, unlike ${\mathrm{\ell }}^2(\mathbb{Z})$, there is no characterization of the cyclicity of $u$ in ${\mathrm{\ell }}^p(\mathbb{Z})$, $1<p<2$, in terms of $\mathcal{Z}(\widehat{u})$ and the divergence of the integral $\int \mathrm{\_}\mathbb{T}\log \mathrm{\mid }\widehat{u}\mathrm{\mid }$. Moreover we give both necessary conditions and sufficient conditions for $u$ to be cyclic in ${\mathrm{\ell }}^p(\mathbb{Z})$, $1<p<2$.