## Abstract

Let*ν* be a signed measure on E^{d} with νE^{d}=0 and ¦ν¦E^{d}<∞. Define*D*
_{s}(ν) as sup ¦*νH*¦ where*H* is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.**Theorem A.** Let*ν* be supported by a finite pointset*p*
_{i}. Then*D*
_{s}(ν)>*c*
_{d}(δ_{1}/*δ*
_{2})^{1/2}{∑_{
i(νp}
_{i})^{2}}^{1/2} where*δ*
_{1} is the minimum distance between two distinct*p*
_{i}, and*δ*
_{2} is the maximum distance. The number*c*
_{d} is an absolute dimensional constant. (The number .05 can be chosen for*c*
_{2} in Theorem A.)**Theorem B.** Let**D** be a disk of unit area in the plane**E**
^{2}, and*p*
_{1},*p*
_{2},...,*p*
_{n} be a set of points lying in**D**. If m if the usual area measure restricted to**D**, while γ*nP*
_{i}=1/*n* defines an atomic measure γ_{n}, then independently of γ_{n},*nD*
_{s}(**m** −*γ*
_{n})≥ .0335*n*
^{1/4}. Theorem B gives an improved solution to the Roth “disk segment problem” as described by Beck and Chen. Recent work by Beck shows that*nD*
_{s}(**m**
*−γ*
_{n})≥*cn*
^{1/4}(log*n*)^{−7/2}.

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Alexander, R. Geometric methods in the study of irregularities of distribution.
*Combinatorica* **10, **115–136 (1990). https://doi.org/10.1007/BF02123006

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### AMS subject classification (1980)

- 52 A 22
- 10 K 30