  We consider the star discrepancy of two-dimensional sequences made up as a
hybrid between a Kronecker sequence and a perturbed Halton sequence in base 2,
where the perturbation is achieved by a digital-sequence construction in the
sense of Niederreiter whose generating matrix contains a periodic perturbing
sequence of a given period length. Under the assumption that the Kronecker
sequence involves a parameter with bounded continued fraction coefficients
sharp discrepancy estimates are obtained. Furthermore, we study the problem
from a metric point of view as well. Finally, we also present sharp general and
tight metric bounds for certain lacunary trigonometric products which appear to
be strongly related to these problems.
