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GEOMETRIC DISCREPANCY THEORY AND UNIFORM DISTRIBUTION
, 2004
"... A sequence s1,s2,... in U =[0,1) is said to be uniformly distributed if, in the limit, the number of sj falling in any given subinterval is proportional to its length. Equivalently, s1,s2,... is uniformly distributed if the sequence of equiweighted atomic probability measures µN(sj) =1/N, supported ..."
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A sequence s1,s2,... in U =[0,1) is said to be uniformly distributed if, in the limit, the number of sj falling in any given subinterval is proportional to its length. Equivalently, s1,s2,... is uniformly distributed if the sequence of equiweighted atomic probability measures µN(sj) =1/N, supported by the initial Nsegments
Geometric Set Systems
, 1998
"... Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higherdimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, e ..."
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Let X be a finite point set in the plane. We consider the set system on X whose sets are all intersections of X with a halfplane. Similarly one can investigate set systems defined on point sets in higherdimensional spaces by other classes of simple geometric figures (simplices, balls, ellipsoids, etc.). It turns out that simple combinatorial properties of such set systems (most notably the VapnikChervonenkis dimension and related concepts of shatter functions) play an important role in several areas of mathematics and theoretical computer science. Here we concentrate on applications in discrepancy theory, in combinatorial geometry, in derandomization of geometric algorithms, and in geometric range searching. We believe that the described tools might be useful in other areas of mathematics too. 1 Introduction For a set system S ` 2 X on an arbitrary ground set X and for A ` X, we write Sj A = fS " A; S 2 Sg for the set system induced by S on A (or the trace of S on A). Let H den...