Probabilistic diophantine approximation and the distribution of Halton-Kronecker sequences

By a Halton-Kronecker sequence we mean a sequence in the s + t-dimensional unit-cube which is the combination of an s-dimensional Halton sequence and a t-dimensional Kronecker sequence ({n . alpha})(n=0,1,...) with alpha is an element of R-t. 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 alpha is an element of R-t for the discrepancy D-N of a Halton-Kronecker sequence we have D-N = O((log N)(s+t+c)/N) for all epsilon > 0, which most probably essentially is the best possible metrical result for this type of sequences. (C) 2013 Elsevier Inc. All rights reserved.