Optimal periodic L 2 -discrepancy and diaphony bounds for higher order digital sequences

We present an explicit construction of infinite sequences of points (x_0, x_1, x_2, …) in the d -dimensional unit-cube whose periodic L 2 -discrepancy satisfies L_2,N^ per({x_0,x_1,…, x_N-1}) ≤ C_d N^-1 (log N)^d/2 for all N ≥ 2, where the factor C_d > 0 depends only on the dimension d . The construction is based on higher order digital sequences as introduced by J. Dick in the year 2008. The result is best possible in the order of magnitude in N according to a Roth-type lower bound shown first by P.D. Proinov. Since the periodic L 2 -discrepancy is equivalent to P. Zinterhof's diaphony the result also applies to this alternative quantitative measure for the irregularity of distribution.