A remark on the geometry of spaces of functions with prime frequencies

For any positive integer r, denote by Pr the set of all integers γ ∈ Z having at most r prime divisors. We show CPr (T), the space of all continuous functions on the circle T whose Fourier spectrum lies in Pr, contains a complemented copy of `1. In particular, CPr (T) is not isomorphic to C(T), nor to the disc algebra A(D). A similar result holds in the L1 setting. For any set of “frequencies” Λ ⊂ Z, let us denote by CΛ(T) the space of all continuous functions f on the circle T whose Fourier spectrum lies in Λ, i.e. f̂(γ) = 0 for every γ ∈ Z\Λ. A classical topic in harmonic analysis is to try to understand how various “thinness” properties of the set Λ are reflected in geometrical properties of the Banach space CΛ(T). For example, by famous results of Varopoulos and Pisier, Λ is a Sidon set if and only of CΛ is somorphic to ` , if and only if CΛ has cotype 2. More generally, one can start with any Banach space X naturally contained in L(T) and consider the spaces XΛ of all functions f ∈ X whose Fourier spectrum lies in Λ. We refer to the monographs [6] or [13] to know more on the various notions of thin sets of integers. We concentrate here on sets Λ ⊂ Z closely related to the set P of prime numbers. Considering the primes as a “thin” set is a question of point of view. On the one hand, P is a small set since it has zero density in Z. On the other hand, the envelopping sieve philosophy as exposed in [11] and [10] says that the primes can be looked at as a subsequence of density of a linear combination of arithmetic sequences, a fact that is brilliantly illustrated in [2] by showing that P contains arbitrarily long arithmetic progressions. From the harmonic analysis standpoint, let us just quote two results going in opposite directions: it was shown by F. LustPiquard [7] that CP(T) contains subspaces isomorphic to c0, a property shared by C(T) and the disc algebra A(D) = CZ+(T), which indicates that P is not a very thin set; but on the other hand ([4]) any f ∈ LP (T) is “totally ergodic” (i.e. fg has a unique invariant mean for any g ∈ C(T)), a property shared by very thin sets such as Sidon sets and not satisfied by Z and Z+. In this note, we show that the geometry of CP(T) is quite different from that of C(T) or the disc algebra A(D) = CZ+(T), and that LP(T) is very different from L(T). Hence, the primes definitely do not behave like an arithmetic progression. Further, this disruption occurs even with the set of integers having at most r primes preprint 2000 Mathematics Subject Classification. 11A41, 42A55, 43A46, 46B03 .