Irregularities of two-colourings of theN×N square lattice


The first theorem of this paper concerns a result which says, in aquantitative form, that a two-colouring of the points of theN×N square lattice cannot be well-distributed simultaneously relative to allline segments. The proof is an adaptation of an analytic method of K. F. Roth.

Second we prove a surprisingly sharp result of the same spirit ontilted rectangles. The deeper part of the proof constitutes an immediate application of the so-called “integral equation method” due to W. M. Schmidt.

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Beck, J. Irregularities of two-colourings of theN×N square lattice. Combinatorica 2, 111–123 (1982).

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AMS subject classification (1980)

  • 10 K 30
  • 05 C 55
  • 10 K 35