Large Discrepancy In Homogenous 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 α≥1 and a set of natural numbers of the form {0, [α], [2α], [3α], . . . , [kα]} 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 on the discrepancy.

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Correspondence to Robert Hochberg.

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Hochberg, R. Large Discrepancy In Homogenous Quasi-Arithmetic Progressions. Combinatorica 26, 47–64 (2006).

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Mathematics Subject Classification (2000):

  • 11B25
  • 11K38
  • 05C15