On the discrepancy of certain sequences mod\, 1

For any set of N points in the unit square, and any point (x, y) ia the square, let S (x, y) denote the number of points of the set which are in the rectangle 0 _ ck (log N) -12, where c is a positive con- stant depending only on 1, and D is the discrepancy of a set of N points in the /c-dimensional unit cube. Roth also showed an example of 2 points in the unit square for which the discrepancy D _< 2n -k 1. In 1960 J. H. Halton