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We study the cyclicity of vectors $u$ in $\ell^p(\mathbb{Z})$. It is knownthat a vector $u$ is cyclic in $\ell^2(\mathbb{Z})$ if and only if the zeroset, $\mathcal{Z}(\widehat{u})$, of its Fourier transform, $\widehat{u}$, hasLebesgue measure zero and $\log |\widehat{u}| \not \in L^1(\mathbb{T})$, where$\mathbb{T}$ is the unit circle. Here we show that, unlike$\ell^2(\mathbb{Z})$, there is no characterization of the cyclicity of $u$ in$\ell^p(\mathbb{Z})$, $1<p<2$, in terms of $\mathcal{Z}(\widehat{u})$ and thedivergence of the integral $\int\_\mathbb{T} \log |\widehat{u}| $. Moreover wegive both necessary conditions and sufficient conditions for $u$ to be cyclicin $\ell^p(\mathbb{Z})$, $1<p<2$.

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