# zbMATH — the first resource for mathematics

Principles of a new method in the study of irregularities of distribution. (English) Zbl 0721.11028
A new method in the theory of irregularities of distribution is presented; the method is mainly based on ideas of integral geometry. Define a signed measure $$\nu =\nu^{+}-\nu^{-}$$ where $$\nu^{+}$$ is the normalized Lebesgue measure (on some region in Euclidean space) and $$\nu^{-}$$ is the atomic measure that assigns weight $$1/n$$ to given points $$p_k$$ for $$k\leq n$$. The problem discussed in the paper is to find lower bounds for the discrepancy
$D(\nu)=\sup | \nu(H)|,$ where $$H$$ ranges over all open half-spaces. The main idea is to consider functionals of the type
$I^\alpha(\nu)=\iint | p - q|^\alpha\, d\nu(p)\,d\nu(q).$ For various choices of $$\alpha$$ and $$\nu$$ the functional $$I^\alpha$$ has had important applications, certainly for $$\alpha <0$$ in potential theory.
The present work is only concerned with $$0<\alpha <2$$ and the following fundamental estimate is proved:
Theorem A. Suppose that $$M$$ is a convex $$t$$-dimensional hypersurface embedded in a Euclidean space $$E^{t+1}$$. Let $$K$$ be any set of unit $$t$$-measure lying on $$M$$ and let $$p_1,\dots,p_n$$ be arbitrary points of $$M$$. Then for $$0<\alpha <2$$
$n^2\, | I^\alpha(\nu)| > c(t,\alpha)n^{1-\alpha /t}, \tag{1}$ where the constant $$c$$ is independent of $$\nu$$, $$M$$ and $$K$$.
For the special case of the $$t$$-sphere J. Beck [Mathematika 31, 33–41 (1984; Zbl 0553.51013)] has established a result that implies (1) by means of Fourier transformation techniques; G. Wagner has recently established a result equivalent to (1) for $$t=2$$ by means of spherical harmonics.
Using ideas from integral geometry the following second main result follows from Theorem A:
Theorem B. Let $$\nu$$ be as in Theorem A and assume that $$\nu$$ is supported by a disk of radius $$r$$. Then
$n\,D(\nu)>c_tR^{-1/2}n^{1/2-1/2d}, \tag{2}$ where the constant $$c_t$$ depends only on $$t$$.
In the case of the $$t$$-sphere (2) was obtained by J. Beck applying his Fourier transform method, a little weaker result is due to W. M. Schmidt by means of his integral equation method; cf. J. Beck and W. W. L. Chen [Irregularities of distribution. Cambridge etc.: Cambridge University Press (1987; Zbl 0617.10039)].

##### MSC:
 11K38 Irregularities of distribution, discrepancy
Full Text:
##### References:
 [1] [A1] Alexander, R.: On the sum of distances betweenn points on a sphere. Acta Math. Hung23, 443-448 (1972) · Zbl 0265.52009 · doi:10.1007/BF01896964 [2] [A2] Alexander, R.: Generalized sums of distances. Pacific. J. Math.56, 297-304 (1975) · Zbl 0306.53058 [3] [A3] Alexander, R.: Geometric methods in the study of irregularities of distribution. Combinatorica10 (1990) (to appear) [4] [A-S] Alexander, R., Stolarsky, K.B.: Extremal problems of distance geometry related to energy integrals. Trans. Am. Soc.193, 1-31 (1973) · Zbl 0293.52005 · doi:10.1090/S0002-9947-1974-0350629-3 [5] [B1] Beck, J.: Sums of distances between points on a sphere ? an application of the theory of irregularities of distribution to discrete geometry. Mathematica31, 33-41 (1984) · Zbl 0553.51013 [6] [B2] Beck, J.: On a problem of K.F. Roth concerning irregularities of point distribution. Invent. Math.74, 477-487 (1983) · Zbl 0528.10037 · doi:10.1007/BF01394247 [7] [BC] Beck, J., Chen, W. W. L.: Irregularities of distribution (Cambridge Tracts in Mathematics, vol. 89). Cambridge: Cambridge University Press 1987 [8] [Bm] Blumenthal, L. M.: Theory and applications of distance geometry. Oxford: Clarendon Press 1953 · Zbl 0050.38502 [9] [E] Erdös, P.: Problems and results on diophantine approximation. Compositio Math.16, 52-66 (1964) [10] [H] Helgason, S.: Groups and geometric analysis. Orlando: Academic Press 1984 · Zbl 0543.58001 [11] [KN] Kuipers, L., Niederreiter, H.: Uniform distribution of sequences. New York: John Wiley 1974 · Zbl 0281.10001 [12] [R] Roth, K.F.: On irregularities of distribution. Mathematica1, 73-79 (1954) · Zbl 0057.28604 [13] [Rg] Rogers, A.D.: Theory and applications of a functional from metric geometry. Ph.D. Thesis University of Illinois: Urbana (1990) [14] [Sa] Santaló, L.A.: Integral geometry and geometric probability (Encyclopedia of Mathematics and its Applications, vol. 1). Reading, Mass.: Addison-Wesley 1976 · Zbl 0342.53049 [15] [Sb] Schoenberg, I.J.: On certain metric spaces arising from Euclidean spaces by change of metric and their embedding in Hilbert space. Ann. Math.38, 787-793 (1937) · Zbl 0017.36101 · doi:10.2307/1968835 [16] [Sm1] Schmidt, W.M.: Irregularities of distribution. IV. Invent. Math.7, 55-82 (1969) · Zbl 0172.06402 · doi:10.1007/BF01418774 [17] [Sm2] Schmidt, W.M.: Irregularities of distribution. VII. Acta Arith.21, 45-50 (1972) · Zbl 0244.10035 [18] [St.] Stolarsky, K.B.: Sums of distances between points on a sphere. II. Proc. Am. Math. Soc.41, 575-582 (1973) · Zbl 0274.52012 · doi:10.1090/S0002-9939-1973-0333995-9 [19] [W] Weyl, H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann.77, 313-352 (1916) · JFM 46.0278.06 · doi:10.1007/BF01475864 [20] [Wg] Wagner, G.: On means of distances on the surface of a sphere I, II. (to appear)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.