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The star discrepancy $D_N^*(\mathcal{P})$ is a quantitative measure for theirregularity of distribution of a finite point set $\mathcal{P}$ in themulti-dimensional unit cube which is intimately related to the integrationerror of quasi-Monte Carlo algorithms. It is known that for every integer $N\ge 2$ there are point sets $\mathcal{P}$ in $[0,1)^d$ with $|\mathcal{P}|=N$and $D_N^*(\mathcal{P}) =O((\log N)^{d-1}/N)$. However, for small $N$ comparedto the dimension $d$ this asymptotically excellent bound is useless (e.g. for$N \le {\rm e}^{d-1}$).  In 2001 it has been shown by Heinrich, Novak, Wasilkowski andWo\'{z}niakowski that for every integer $N \ge 2$ there exist point sets$\mathcal{P}$ in $[0,1)^d$ with $|\mathcal{P}|=N$ and $D_N^*(\mathcal{P}) \le C\sqrt{d/N}$. Although not optimal in an asymptotic sense in $N$, this upperbound has a much better (and even optimal) dependence on the dimension $d$.  Unfortunately the result by Heinrich et al. and also later variants thereofby other authors are pure existence results and until now no explicitconstruction of point sets with the above properties are known. Quite recentlyL\"obbe studied lacunary subsequences of Kronecker's $(n\boldsymbol{\alpha})$-sequence and showed a metrical discrepancy bound of theform $C \sqrt{d (\log d)/N}$ with implied absolute constant $C>0$ independentof $N$ and $d$.  In this paper we show a corresponding result for digital Kronecker sequences,which are a non-archimedean analog of classical Kronecker sequences.

Metrical star discrepancy bounds for lacunary subsequences of digital Kronecker-sequences and polynomial tractability