  A duality formula, of the Hardy and Littlewood type for multidimensional
Gaussian sums, is proved in order to estimate the asymptotic long time behavior
of distribution of Birkhoff sums $S_n$ of a sequence generated by a skew
product dynamical system on the $\mathbb{T}^2$ torus, with zero Lyapounov
exponents. The sequence, taking the values $\pm 1$, is pairwise independent
(but not independent) ergodic sequence with infinite range dependence. The
model corresponds to the motion of a particle on an infinite cylinder, hopping
backward and forward along its axis, with a transversal acceleration parameter
$\alpha$. We show that when the parameter $\alpha /\pi$ is rational then all
the moments of the normalized sums $E((S_n/\sqrt{n})^k)$, but the second, are
unbounded with respect to n, while for irrational $\alpha /\pi$, with bounded
continuous fraction representation, all these moments are finite and bounded
with respect to n.
