Probabilistic diophantine approximation and the distribution of Halton–Kronecker sequences

By a Halton–Kronecker sequence we mean a sequence in the s+ts+t-dimensional unit-cube which is the combination of an ss-dimensional Halton sequence and a tt-dimensional Kronecker sequence ({n⋅α})n=0,1,… with α∈Rt. The investigation of such ‘hybrid sequences’ for their use in Monte Carlo and quasi-Monte Carlo methods first was motivated by Spanier (1995) [20]. By suitably adapting techniques of Jozsef Beck on probabilistic diophantine approximation, developed in Beck (1994) [2], we can show that for almost all α∈Rt for the discrepancy DNDN of a Halton–Kronecker sequence we have DN=O((logN)s+t+ϵN) for all ϵ>0ϵ>0, which most probably essentially is the best possible metrical result for this type of sequences.