Covering numbers, dyadic chaining and discrepancy

  • Authors:

  • Christoph Aistleitner

  • Affiliations:

  • -

  • Venue:
  • Journal of Complexity
  • Year:
  • 2011

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Abstract

In 2001 Heinrich, Novak, Wasilkowski and Wozniakowski proved that for every s=1 and N=1 there exists a sequence (z"1,...,z"N) of elements of the s-dimensional unit cube such that the star-discrepancy D"N^* of this sequence satisfies D"N^*(z"1,...,z"N)@?csN for some constant c independent of s and N. Their proof uses deep results from probability theory and combinatorics, and does not provide a concrete value for the constant c. In this paper we give a new simple proof of this result, and show that we can choose c=10. Our proof combines Gnewuch's upper bound for covering numbers, Bernstein's inequality and a dyadic partitioning technique.