On irregularities of distribution, III

On Irregularities of Distribution Iii

Dedicated to the memory of Gerold Wagner Abstract We study the L w -norm (2 < W < oo) of the discrepancy of a sequence of points in the unit cube relative to similar copies of a given convex polygon. In particular, we prove the rather surprising result that the estimates obtained have the same order of magnitude as the analogous question when the sequence of points is replaced by a set of point...

On Irregularities of Distribution

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Note on Irregularities of Distribution

Irregularities of Point Distribution Relative to Convex Polygons Iii

The situation is somewhat different when the aligned rectangles are replaced by similar copies of a given convex polygon. More precisely, suppose that 0 is a distribution of N points in the unit square U ̄ [0, 1]#, treated as a torus. Suppose that AXU is a closed convex polygon of diameter less than 1 and with centre of gravity at the origin O. For every real number r satisfying 0% r% 1 and ever...

A note on irregularities of distribution

Let d ≥ 0 be a fixed integer. Suppose X = (x1, x2, x3, . . .) is a sequence in [0, 1) with the property that for every n ≤ N , each of the intervals [ k−1 n , k n ), 1 ≤ k ≤ n, contains at least one xi with 1 ≤ i ≤ n+d. We show that this implies that N = O(d). This is a generalization of a question raised by Steinhaus some 50 years ago.