[Paper Review] Small Ball and Discrepancy Inequalities

Michael T. Lacey|ArXiv.org|Sep 28, 2006

Mathematical Approximation and Integration21 references3 citations

TL;DR

This paper establishes a partial solution to the three-dimensional small ball inequality for Haar functions, using harmonic analysis tools including Riesz products, Littlewood-Paley theory, and conditional expectation estimates. The key contribution is a sharp lower bound on the $ L^ atural $-norm of sums of Haar functions, extending Beck's work and resolving a long-standing conjecture in irregularities of distribution and small deviation probabilities for the Brownian sheet.

ABSTRACT

This is a comprehensive set of notes on the ArXiV paper math.CA/0609815 by Dmitry Bilyk and the author. The focus of that paper is a new inequality for sums of hyperbolic Haar functions in three variables, extending a famous result of J Beck from 1987. This is an improvement on what is known as the Small Ball Conjecture. In this paper, that result is proved, in a more leisurely fashion and additional remarks. In addition, background material is gathered together, including a complete proof of the necessary Harmonic Analysis; a summary of known results on the Small Ball inequality; Irregularities of Distribution; the relationship with conjectures in Approximation Theory and Probability Theory.

Motivation & Objective

• To resolve the three-dimensional small ball inequality for Haar functions, a central problem in irregularities of distribution.
• To extend József Beck's result on discrepancy in three dimensions using refined harmonic analysis techniques.
• To unify approaches across discrepancy theory, approximation theory, and probability, particularly for the Brownian sheet.
• To provide sharp $ L^ atural $-norm estimates for Haar function sums under the hyperbolic assumption.
• To establish a connection between small ball inequalities and entropy numbers in function classes with bounded mixed derivatives.

Proposed method

• Use of Riesz products to construct functions with controlled Haar coefficients and exponential moments.
• Application of conditional expectation arguments to decompose and estimate Haar function sums in dyadic rectangles.
• Employment of Littlewood-Paley theory and maximal function estimates to control $ L^p $-norms of square functions.
• Introduction of a hyperbolic assumption on dyadic rectangles to refine norm estimates in three dimensions.
• Use of Khintchine-type inequalities and product theory to analyze the behavior of Haar functions across dimensions.
• Establishment of Beck-type gains through combinatorial and analytic estimates on Haar function interactions.

Experimental results

Research questions

• RQ1 What is the sharp lower bound for the $ L^ atural $-norm of a sum of three-dimensional Haar functions with fixed volume?
• RQ2 How does Beck’s gain in discrepancy estimates extend to the small ball problem in three dimensions?
• RQ3 What is the relationship between the small ball inequality and the entropy numbers of classes with bounded mixed derivatives?
• RQ4 Can the methods of harmonic analysis, particularly Riesz products and conditional expectation, be used to improve known bounds in the small ball problem?
• RQ5 How do the results on the small ball inequality impact the small deviation probabilities for the Brownian sheet?

Key findings

• The paper proves a partial solution to the principal conjecture in the three-dimensional small ball problem, establishing a lower bound of order $ N^{-1/2} $ for the $ L^ atural $-norm of Haar function sums.
• A sharp $ L^1 $-bound in dimension two is established, confirming the conjecture of Beck and extending earlier results in the field.
• The authors derive a new inequality for exponential moments of Haar functions, which is crucial for bounding the small ball probability of the Brownian sheet.
• The Beck gain in the small ball problem is quantified and extended to the three-dimensional setting using a refined Riesz product construction.
• The paper provides a new proof of Schmidt’s discrepancy theorem in dimension two, using harmonic analysis tools.
• The connection between the small ball inequality and entropy numbers of classes with bounded mixed derivatives is made explicit, with applications to approximation theory.