A Remark on The Geometry of Spaces of Functions with Prime Frequencies

For any positive integer r, denote by ({mathcal{P}_{r}}) the set of all integers ({gamma in mathbb{Z}}) having at most r prime divisors. We show that ({C_{mathcal{P}_{r}}(mathbb{T})}) , the space of all continuous functions on the circle ({mathbb{T}}) whose Fourier spectrum lies in ({mathcal{P}_{r}}) , contains a complemented copy of ({ell^{1}}) . In particular, ({C_{mathcal{P}_{r}}(mathbb{T})}) is not isomorphic to ({C(mathbb{T})}) , nor to the disc algebra ({A(mathbb{D})}) . A similar result holds in the L ¹ setting.