Resumen de Pair correlations of the leVeque sequence on the polydisc

Ramachandran D. Nair

• We consider a system of �forms� defined for ? = (zij) on a subset of by where d = d1 + + dl and for each pair of integers (i,j) with 1 = i = l, 1 = j = di we denote by a strictly increasing sequence of natural numbers. Let = {z : |z| < 1} and let where for each pair (i, j) we have Xij = . We study the distribution of the sequence on the l-polydisc defined by the coordinatewise polar fractional parts of the sequence Xk(?) = (L1(?)(k),. . ., Ll(?)(k)) for typical ? in More precisely for arcs I1, . . ., I2l in , let B = I1 × × I2l be a box in and for each N = 1 define a pair correlation function by and a discrepancy by ?N = {VN(B) - N(N-1)leb(B)}, where the supremum is over all boxes in . We show, subject to a non-resonance condition on , that given e > 0 we have ?N = o(N(log log N)1+e) for almost every . Similar results on extremal discrepancy are also proved. Our results complement those of I. Berkes, W. Philipp, M. Pollicott, Z. Rudnick, P. Sarnak, R Tichy and the author in the real setting.