Combinatorica

, Volume 10, Issue 2, pp 115–136

# Geometric methods in the study of irregularities of distribution

• R. Alexander
Article

## Abstract

Letν be a signed measure on E d with νE d =0 and ¦ν¦Ed<∞. DefineDs(ν) as sup ¦νH¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Letν be supported by a finite pointsetpi. ThenDs(ν)>cd1/δ2)1/2{∑i(νpi)2}1/2 whereδ1 is the minimum distance between two distinctpi, andδ2 is the maximum distance. The numbercd is an absolute dimensional constant. (The number .05 can be chosen forc2 in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE2, andp1,p2,...,pn be a set of points lying inD. If m if the usual area measure restricted toD, while γnPi=1/n defines an atomic measure γn, then independently of γn,nDs(mγn)≥ .0335n1/4. Theorem B gives an improved solution to the Roth “disk segment problem” as described by Beck and Chen. Recent work by Beck shows thatnDs(m−γn)≥cn1/4(logn)−7/2.

52 A 22 10 K 30

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