  Consider the plane as a checkerboard, with each unit square colored black or
white in an arbitrary manner. In a previous paper we showed that for any such
coloring there are straight line segments, of arbitrarily large length, such
that the difference of their white length minus their black length, in absolute
value, is at least the square root of their length, up to a multiplicative
constant. For the corresponding "finite" problem ($N \times N$ checkerboard) we
had proved that we can color it in such a way that the above quantity is at
most $C \sqrt{N \log N}$, for any placement of the line segment. In this
followup we show that it is possible to color the infinite checkerboard with
two colors so that for any line segment $I$ the excess of one color over
another is bounded above by $C_\epsilon \Abs{I}^{\frac12+\epsilon}$, for any
$\epsilon>0$. We also prove lower bounds for the discrepancy of circular arcs.
Finally, we make some observations regarding the $L^p$ discrepancies for
segments and arcs, $p<2$, for which our $L^2$-based methods fail to give any
reasonable estimates.

Classical Analysis and ODEs;Number Theory