Estimates of the Discrepancy Function in Exponential Orlicz Spaces

We prove that in all dimensions n at least 3, for every integer N there exists a distribution of points of cardinality $N$, for which the associated discrepancy function D_N satisfies the estimate an estimate, of sharp growth rate in N, in the exponential Orlicz class exp)L^{2/(n+1)}. This has recently been proved by M.~Skriganov, using random digit shifts of binary digital nets, building upon the remarkable examples of W.L.~Chen and M.~Skriganov. Our approach, developed independently, complements that of Skriganov.