  The 2-colouring discrepancy of arithmetic progressions is a well-known
problem in combinatorial discrepancy theory. In 1964, Roth proved that if each
integer from 0 to N is coloured red or blue, there is some arithmetic
progression in which the number of reds and the number of blues differ by at
least (1/20) N^{1/4}. In 1996, Matousek and Spencer showed that this estimate
is sharp up to a constant. The analogous question for homogeneous arithmetic
progressions (i.e., the ones containing 0) was raised by Erdos in the 1930s,
and it is still not known whether the discrepancy is unbounded. However, it is
easy to construct partial colourings with density arbitrarily close to 1 such
that all homogeneous arithmetic progressions have bounded discrepancy.
  A related problem concerns the discrepancy of quasi-progressions. A
quasi-progression consists of successive multiples of a real number, with each
multiple rounded down to the nearest integer. In 1986, Beck showed that given
any 2-colouring, the quasi-progressions corresponding to almost all real
numbers in (1, \infty) have discrepancy at least log* N, the inverse of the
tower function. We improve the lower bound to (log N)^{1/4 - o(1)}, and also
show that there is some quasi-progression with discrepancy at least (1/50)
N^{1/6}. Our results remain valid even if the 2-colouring is replaced by a
partial colouring of positive density.
