Irregularities for distribution IX

On Irregularities of Distribution

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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 ...

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

Upper bounds in irregularities of distribution

The subject of irregularities of distribution arises from uniform distribution, but is of independent interest, and owes its current prominance to the fundamental contribution of K.F. Roth [23–27] and W.M. Schmidt [28–38]. While the theory of uniform distribution may be described as qualitative, the theory of irregularities of distribution is definitely quantitative in nature, as one seeks to m...