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A duality formula, of the Hardy and Littlewood type for multidimensionalGaussian sums, is proved in order to estimate the asymptotic long time behaviorof distribution of Birkhoff sums $S_n$ of a sequence generated by a skewproduct dynamical system on the $\mathbb{T}^2$ torus, with zero Lyapounovexponents. The sequence, taking the values $\pm 1$, is pairwise independent(but not independent) ergodic sequence with infinite range dependence. Themodel corresponds to the motion of a particle on an infinite cylinder, hoppingbackward and forward along its axis, with a transversal acceleration parameter$\alpha$. We show that when the parameter $\alpha /\pi$ is rational then allthe moments of the normalized sums $E((S_n/\sqrt{n})^k)$, but the second, areunbounded with respect to n, while for irrational $\alpha /\pi$, with boundedcontinuous fraction representation, all these moments are finite and boundedwith respect to n.Comment: To be published in J. Phys.

Multidimensional Gaussian sums arising from the distribution of Birkhoff sums in zero entropy dynamical systems