Titlebook: Geometric Discrepancy; An Illustrated Guide Jiří Matoušek Book 1999 Springer-Verlag Berlin Heidelberg 1999 Combinatorics.Dimension.Diskrepa

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978-3-642-03941-6Springer-Verlag Berlin Heidelberg 1999

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https://doi.org/10.1007/978-981-10-7500-1In this chapter, we are going to investigate the combinatorial discrepancy, an exciting and significant subject in its own right. From Section 1.3, we recall the basic definition: If . is a finite set and . ⊑ 2. is a family of sets on .,a . is any mapping ., and we have disc ., where .

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https://doi.org/10.1007/978-3-658-18971-6id, placed in the unit square in an appropriate scale, as in Fig. 2.1(a). It is easy to see that this gives discrepancy of the order .. Another attempt might be n independent random points in the unit square as in Fig. 2.1(b), but these typically have discrepancy about . as well. (In fact, with high

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https://doi.org/10.1007/978-3-476-05622-1seen some lower bounds in Chapter 4 but not in a geometric setting). So far we have not answered the basic question, Problem 1.1, namely whether the discrepancy for axis-parallel rectangles must grow to infinity as n . → ∞. An answer is given in Section 6.1, where we prove that .(.,..) is at least o