On irregularities of distribution

On Irregularities of Distribution

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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...

Note on Irregularities of Distribution

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.

Irregularities of Distribution, Ii

(1.2) \D[zu ..., zn ; B(x, r)]\ > c2(K, e). n < Note that the exponent {\ — (1/2K) e) of n is essentially the best possible: (2 — (1/2K) e) cannot be replaced by the exponent (2 — (1/2K) + e) with e > 0. Note further that (1.1) or (1.2) above guarantees the existence of a ball in U with 'error' very large as compared to that of boxes in U with sides parallel to the axes. We recall that in 1954 ...