The exact order of generalized diaphony and multidimensional numerical integration

For a point set in the multidimensional unit torus we introduce an L-kappa- measure of uniformity of distribution, which for kappa = 2 reduces to diaph ony (and thus in this case essentially coincides with Weyl L-2-discrepancy) . For kappa is an element of [1, 2] we establish a sharp asymptotic for thi s new measure as the number of points of the set tends to infinity. Upper a nd lower-bound estimates are given also for kappa > 2.