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The Small Ball Inequality in all Dimensions
by
Michael T Lacey
Georgia Institute of Technology
Coauthors: Dmitriy Bilyk and Armen Vagharshakyan
The Small Ball Inequality concerns a lower bound on the L∞ norm of sums of Haar functions adapted to rectangles of a fixed volume. The relevant conjecture is improvement of the average case lower bound by an amount that is the square-root log of the volume of the rectangles. We obtain the first non-trivial improvement over the average case bound in dimensions four and higher. The conjecture is known in dimension 2, a result due to Wolfgang Schmidt and Michel Talagrand, with important contributions from Halasz and Temlyakov. Jozef Beck established a prior result in three dimensions, which argument we extend and simplify.
This question is related to (1) Irregularities of Distribution, (2) Probability and (3) Approximation Theory.
Date received: January 14, 2008
Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavq-09.