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

A duality formula, of the Hardy and Littlewood type for multidimensional Gaussian sums, is proved in order to estimate the asymptotic long-time behaviour of the distribution of the Birkhoff sums Sn of a sequence generated by a skew product dynamical system on the T2 torus, with zero Lyapounov exponents. The sequence, taking the values ±1, is a pairwise independent (but not independent otherwise) 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 α. We show that when the parameter α/π is rational then all the moments of the normalized sums E((Sn/√n)k), but the second, are unbounded with respect to n, while for irrational α/π, with bounded continuous fraction representation, all these moments are finite and bounded with respect to n