Recurrent and almost-periodic sequences

A sequence g : N → C is called almost-periodic if it belongs to the completion of the C-linear space spanned by the sequences e _ϑ with ϑ ∈ R/Z, where e _ϑ(n) = e ^2πiϑn for n ∈ N, under the semi-norm . Every has a mean value . A sequence g: N → C is called recurrent if it satisfies a linear recurrence equation of the form with coefficients a _k−1...,a ₀ ∈ C, a ₀ ≠ 0, and with some numbers k, n ₀ ∈ N ∪ {0}. Let ℜ denote the space of recurrent sequences. It is shown that a sequence cannot belong to ℜ if M (g e _ϑ) ≠ 0 for infinitely many ϑ ∈ R/Z, which extends a recent result of Spilker. The proof is based on Kronecker’s rationality test.