GEOMETRIC METHODS IN THE STUDY OF IRREGULARITIES OF DISTRIBUTION

Let nu be a signed measure on E(d) with nu-E(d) = 0 and \nu\E(d) < infinity. Define D(s)(nu) as sup \nu-H\ 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 nu be supported by a finite pointset p(i). Then D(s)(nu) > c(d)(delta-1/delta-2)1/2 {SIGMA-i(nu-p(i))2}1/2, where delta-1 is the minimum distance between two distinct p(i), and delta-2 is the maximum distance. The number c(d) is an absolute dimensional constant. (The number .05 can be chosen for c2 in Theorem A.) Theorem B. Let D be a disk of unit area in the plane E2, and p1, p2, ..., p(n) be a set of points lying in D. If m if the usual area measure restricted to D, while gamma-np(i) = 1/n defines an atomic measure gamma-n, then independently of gamma-n, nD(s)(m-gamma-n) greater-than-or-equal-to .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 that nD(s)(m-gamma-n) greater-than-or-equal-to cn1/4(log n)-7/2.