Cyclicity in ℓp spaces and zero sets of the Fourier transforms

We study the cyclicity of vectors u in ℓp(Z). It is known that a vector u is cyclic in ℓ2(Z) if and only if the zero set, Z(uˆ), of its Fourier transform, uˆ, has Lebesgue measure zero and log⁡|uˆ|∉L1(T), where T is the unit circle. Here we show that, unlike ℓ2(Z), there is no characterization of the cyclicity of u in ℓp(Z), 1