On the small ball inequality in three dimensions

Dmitriy Bilyk, Michael T. Lacey

Research output: Contribution to journalArticlepeer-review

39 Scopus citations


Let hR denote an L-normalized Haar function adapted to a dyadic rectangle R ⊂ [0, 1]3. We show that there is a positive η < 1/2 so that for all integers n and coefficients α(R), we have (Equation Presented) This is an improvement over the trivial estimate by an amount of n, while the small ball conjecture says that the inequality should hold with η = 1/2. There is a corresponding lower bound on the L-norm of the discrepancy function of an arbitrary distribution of a finite number of points in the unit cube in three dimensions. The prior result, in dimension three, is that of József Beck [1, Theorem 1.2], in which the improvement over the trivial estimate was logarithmic in n. We find several simplifications and extensions of Beck's argument to prove the result above.

Original languageEnglish (US)
Pages (from-to)81-115
Number of pages35
JournalDuke Mathematical Journal
Issue number1
StatePublished - May 15 2008


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