Large discrepancy in homogeneous quasi-arithmetic progressions

We prove that the class of homogeneous quasi-arithmetic progressions has unbounded discrepancy. That is, we show that given any 2-coloring of the natural numbers and any positive integer D, one can find a real number alpha >= 1 and a set of natural numbers of the form {0, [alpha],[2 alpha],[3 alpha],..., [k alpha]} so that, one color appears at least D times more than the other color. This was already proved by Beck in 1983, but the proof given here is somewhat simpler and gives a better bound oil the discrepancy.