Sharp General and Metric Bounds for the Star Discrepancy of Perturbed Halton–kronecker Sequences

We consider distribution properties of two-dimensional hybrid sequences (zk)k in the unit square of the form zk = ({kα}, xk), where α ∈ (0, 1) is irrational and (xk)k denotes a digital Niederreiter sequence. By definition, the construction of the sequence (xk)k relies on an infinite matrix C with entries in {0, 1}. Two special cases of such matrices were studied by Niederreiter (2009) and by Aistleitner, Hofer, and Larcher (2016). On the one hand, by taking the identity in place of C (so-called Halton–Kronecker sequence), (zk)k satisfies an optimal metric discrepancy estimate. On the other hand, taking C as the identity again, but replacing its first row with an infinite sequence of 1’s (related to the so-called Evil Kronecker sequence) worsens the metrical behavior of (zk)k significantly. We are interested in what happens in between these two cases. In particular, we consider Halton–Kronecker sequences, where the first row of C is exchanged by a periodic perturbation (ck)k of blocks of length n of the form (1, 0, . . . , 0). We give sharp bounds for the star discrepancy of (zk)k in the case where α has bounded continued fraction coefficients, which surprisingly worsen with a decreasing density of 1’s in (ck)k. Furthermore, we show tight metric discrepancy bounds for these sequences, which are in line with the previously known results, i.e., the exponents of our estimates approach the optimal value for n→∞. Moreover, we present sharp general as well as tight metric bounds for lacunary trigonometric products of the form ∏r−1 j=0 ∣∣cos (2jαπ + cjπ/2)∣∣ as a side product to our discrepancy estimates.