# Geometric methods in the study of irregularities of distribution

- 64 Downloads
- 24 Citations

## 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}.

### AMS subject classification (1980)

52 A 22 10 K 30## Preview

Unable to display preview. Download preview PDF.

### References

- [1]R. Alexander: On the sum of distances between
*n*points on a sphere,*Acta Math. Acad. Sci. Hungar.*,**23**(1972), 443–448.CrossRefGoogle Scholar - [2]
- [3]R. Alexander: On the sum of distances between
*n*points on a sphere. II,*Acta Math. Acad. Sci. Hungar.*,**29**(1977), 317–320.CrossRefGoogle Scholar - [4]R. Alexander: Metric averaging in Euclidean and Hilbert spaces,
*Pacific Jour. Math.*,**85**(1979), 1–9.Google Scholar - [5]R. Alexander, andK. B. Stolarsky: Extremal problems of distance geometry related to energy integrals,
*Trans. Amer. Soc.*,**193**(1973), 1–31.Google Scholar - [6]J. Beck: Sums of distances between points on a sphere — an application of the theory of irregularities of distribution to discrete geometry,
*Mathematica*,**31**(1984), 33–41.Google Scholar - [7]J. Beck, andW. W. L. Chen:
*Irregularities of distribution*, Cambridge Tracts in Mathematics,**89**, Cambridge University Press, Cambridge, 1987.Google Scholar - [8]L. Kuipers, andH. Niederreiter:
*Uniform distribution of sequences*, John Wiley, New York,**1974**.Google Scholar - [9]
- [10]L. A. Santaló:
*Integral geometry and geometric probability*, Encyclopedia of Mathematics and its Applications,**1**, Addison-Wesley, Reading, Mass., 1976.Google Scholar - [11]I. J. Schoenberg: On certain metric spaces arising from Euclidean spaces by change of metric and their embedding in Hilbert space,
*Ann. of Math.*,**38**(1937), 787–793.MathSciNetGoogle Scholar - [12]W. M. Schmidt: Irregularities of distribution. IV,
*Invent. Math.*,**7**(1969), 55–82.CrossRefGoogle Scholar - [13]
- [14]K. B. Stolarsky: Sums of distances between points on a sphere,
*Proc. Amer. Math Soc.*,**35**(1972), 547–549.Google Scholar - [15]K. B. Stolarsky: Sums of distances between points on a sphere. II,
*Proc. Amer. Math. Soc.*,**41**(1973), 575–582.Google Scholar - [16]K. B. Stolarsky: Spherical distributions of
*n*points with maximal distance sums are well spaced,*Proc. Amer. Math. Soc.*,**48**(1975), 203–206.Google Scholar - [17]K. B.Stolarsky:
*Discrepancy and sums of distances between points of a metric space*, The geometry of metric and linear spaces, Ed. L. M. Kelly, Springer-Verlag,**1975**, 44–55.Google Scholar - [18]H. Weyl: Über die Gleichverteilung von Zahlen mod. Eins,
*Math. Ann.*,**77**(1916), 313–352.CrossRefMathSciNetGoogle Scholar