  In a series of papers recently "checkerboard discrepancy" has been
introduced, where a black-and-white checkerboard background induces a coloring
on any curve, and thus a discrepancy, i.e., the difference of the length of the
curve colored white and the length colored black. Mainly straight lines and
circles have been studied and the general situation is that, no matter what the
background coloring, there is always a curve in the family studied whose
discrepancy is at least of the order of the square root of the length of the
curve.
  In this paper we generalize the shape of the background, keeping the lattice
structure. Our background now consists of lattice copies of any bounded
fundamental domain of the lattice, and not necessarily of squares, as was the
case in the previous papers. As the decay properties of the Fourier Transform
of the indicator function of the square were strongly used before, we now have
to use a quite different proof, in which the tiling and spectral properties of
the fundamental domain play a role.
