On the Discrepancy for Cartesian Products |
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Authors: | Matousek Jiri |
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Institution: | Department of Applied Mathematics, Charles University Malostranské nám. 25, 118 00 Praha 1, Czech Republic, matousek{at}kam.mff.cuni.cz |
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Abstract: | Let B2 denote the family of all circular discs in the plane.It is proved that the discrepancy for the family {B1 x B2 :B1, B2 B2} in R4 is O(n1/4+) for an arbitrarily small constant > 0, that is, it is essentially the same as that for thefamily B2 itself. The result is established for the combinatorialdiscrepancy, and consequently it holds for the discrepancy withrespect to the Lebesgue measure as well. This answers a questionof Beck and Chen. More generally, we prove an upper bound forthe discrepancy for a family {ki=1Ai:AiAi, i = 1, 2, ..., k},where each Ai is a family in Rdi, each of whose sets is describedby a bounded number of polynomial inequalities of bounded degree.The resulting discrepancy bound is determined by the worstof the families Ai, and it depends on the existence of certaindecompositions into constant-complexity cells for arrangementsof surfaces bounding the sets of Ai. The proof uses Beck's partialcoloring method and decomposition techniques developed for therange-searching problem in computational geometry. |
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