The star discrepancy is a quantitative measure for the irregularity of distribution of a finite point set 𝒫 in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer N ≥ 2 there are point sets 𝒫 in [0, 1)d with |𝒫| = N and . However, for small N compared to the dimension d this asymptotically excellent bound is useless (e.g., for N ≤ ed−1).
In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer N ≥ 2there exist point sets 𝒫 in [0, 1)d with |𝒫| = N and . Although not optimal in an asymptotic sense in N, this upper bound has a much better (and even optimal) dependence on the dimension d.
Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker’s (nα)-sequence and showed a metrical discrepancy bound of the form with implied absolute constant C> 0 independent of N and d.
In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.
 AISTLEITNER, CH.: Covering numbers, dyadic chaining and discrepancy, J. Complexity 27 (2011), 531–540.
 AISTLEITNER, CH.: On the inverse of the discrepancy for infinite dimensional infinite sequences, J. Complexity 29 (2013), 182–194.
 BECK, J.: Probabilistic diophantine approximation, I. Kronecker-sequences, Ann. Math. 140 (1994), 451–502.
 BERNSTEIN, S. N.: The Theory of Probabilities. Gastehizdat Publishing House, Moscow, 1946.
 BILYK, D.—LACEY, M. T. — VAGHARSHAKYAN, A.: On the small ball inequality in all dimensions, J. Funct. Anal. 254 (2008), 2470–2502.
 BOREL,É.: Les probabilités denombrables et leurs applications arithmétiques, Rend. Circ. Mat. Palermo 27 (1909), 247–271.
 DICK, J.: A note on the existence of sequences with small star discrepancy, J. Complexity 23 (2007), 649–652.
 DICK, J.—KUO, F. Y.—SLOAN, I. H.: High-dimensional integration: the quasi-Monte Carlo way, Acta Numer. 22 (2013), 133–288.
 DICK, J.—PILLICHSHAMMER, F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010.
 DOERR, B.—GNEWUCH, M.—SRIVASTAV, A.: Bounds and constructions for the star-discrepancy via δ-covers, J. Complexity 21 (2005), 691–709.
 DRMOTA, M.—TICHY, R. F.: Sequences, Discrepancies and Applications. In: Lecture Notes in Math. Vol. 1651, Springer-Verlag, Berlin, 1997.
 GNEWUCH, M.: Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24 (2008), 154–172.
 GNEWUCH, M.: Gnewuch, M. Entropy, randomization, derandomization, and discrepancy. In: Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proc. Math. Stat. Vol. 23, Springer-Verlag, Heidelberg, 2012, pp. 43–78,
 HEINRICH, S.—NOVAK, E.—WASILKOWSKI, G. W.—WOŹNIAKOWSKI, H.: The inverse of the star discrepancy depends linearly on the dimension, Acta Arith. 96 (2001), 279–302.
 HINRICHS, A.: Covering numbers, Vapnik-Červonenkis classes and bounds on the star-discrepancy, J. Complexity 20 (2004), 477–483.
 KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. John Wiley, New York (1974). Reprint, Dover Publications, Mineola, NY, 2006.
 LARCHER, G.: On the distribution of an analog to classical Kronecker-sequences, J. Number Theory 52 (1995), 198–215.