Recurrent and almost-periodic sequences

A sequence g: N --> C is called almost-periodic if it belongs to the completion A(1) of the C-linear space spanned by the sequences e(theta) with theta is an element of R/Z, where e(theta)(n) = e(2 pi i theta n) for n is an element of N, under the semi-norm \\g\\(1) = (x-->infinity)lim sup 1/x (n less than or equal to x)Sigma \g(n)\. Every g is an element of A(1) has a mean value M(g) = (x --> infinity)lim 1/x n less than or equal to x Sigma g(n). A sequence g: N --> C is called recurrent if it satisfies a linear recurrence equation of the form g(n + k) + a(k - 1)g(n + k - 1) + ... + a(0)g(n) = 0 (n is an element of N, n > n(0)) with coefficients a(k - 1),..., a(0) is an element of C, a(0) not equal 0, and with some numbers k, n(0) is an element of N boolean OR {0}. Let R denote the space of recurrent sequences. It is shown that a sequence g is an element of A(1) cannot belong to R if M(ge(theta) not equal 0 for infinitely many theta is an element of R/Z, which extends a recent result of Spilker. The proof is based on Kronecker's rationality test.