Dedicated to Thomas Ransford on the occasion of his 60th birthday.Abstract. We study cyclicity in the Dirichlet type spaces for outer functions whose zero set is countable.2000 Mathematics Subject Classification. Primary 46E22; Secondary 31A05, 31A15, 31A20, 47B32.

We study the cyclic vectors and the spanning set of the circle for the p β (Z) spaces of all sequences u = u n n∈Z such that u n (1 + |n|) β n∈Z ∈ p (Z) with p > 1 and β > 0. By duality the spanning set is the uniqueness set of the distribution on the circle whose Fourier coefficients are in q −β (Z) where q is the conjugate of p. Our characterizations are given in terms of the Hausdorff dimension and capacity.

Abstract. We study the cyclicity of vectors u in ℓ p (Z). It is known that a vector u is cyclic in ℓ 2 (Z) if and only if the zero set, Z( u), of its Fourier transform, u, has Lebesgue measure zero and log | u| ∈ L 1 (T), where T is the unit circle. Here we show that, unlike ℓ 2 (Z), there is no characterization of the cyclicity of u in ℓ p (Z), 1 < p < 2, in terms of Z( u) and the divergence of the integral T log | u|. Moreover we give both necessary conditions and sufficient conditions for u to be cyclic in ℓ p (Z), 1 < p < 2.