On the small ball inequality in three dimensions

Abstract

Abstract. Let h R 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 This is an improvement over the 'trivial' estimate by an amount of n −η , 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 3, is that of József Beck [1], 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. The Principal Conjecture and the Main ResultsIn one dimension, the class of dyadic intervals in the unit interval is D ≔ {[j2Each dyadic interval has a left and right half, which are also dyadic. Define the Haar functionsNote that we use an L ∞ normalization of these functions, which will make some formulas seem odd to a reader accustomed to the L 2 normalization. In dimension d, a dyadic rectangle in the unit cube [0,1] d is a product of dyadic intervals, thus an element of D d . A Haar function associated to R is defined as a product of the Haar functions associated with each side of R, namelyThis is the usual 'tensor' definition.We will concentrate on rectangles with fixed volume. This is the 'hyperbolic' assumption, that pervades the subject. Our concern is the following Theorem and Conjecture concerning a lower bound on the L ∞ norm of sums of hyperbolic Here, the sum on the right is taken over all rectangles with area at least 2 −n .1.3. Small Ball Conjecture. For dimension d ≥ 3 we have the inequalityThis conjecture is, by one square root of n, better than the trivial estimate available from the Cauchy-Schwartz inequality, see § 2. As well, see that section for an explanation as to why the conjecture is sharp. The case of d = 2 (with a sum over |R| = 2 −n on the right-hand side) was resolved by Talagrand [13]. Temlyakov has given an easier proof of the inequality in its present form [14], [16], which resonates with the ideas of Roth [9], Schmidt [10], and Halász [6].Perhaps, it is worthwhile to explain the nomenclature 'Small Ball' at this point. The name comes from the probability theory. Assume that X t : T → R is a canonical Gaussian process indexed by a set T. The Small Ball Problem is concerned with estimates of P(sup t∈T |X t | < ε) as ε goes to zero, i.e the probability that the random process takes values in an L ∞ ball of small radius. The reader is advised to consult a paper by Kuelbs and Li In dimension d = 2, this conjecture has been resolved by Talagrand in the already cited paper [13], in which he used a version of (1.2) for continuous wavelets in place of Haars to prove the lower bound in the inequality above. In higher dimensions, the upper bounds are established and the known lower bounds miss the conjecture by a single power of the logarit...

On the Small Ball Inequality in all dimensions

“…In dimension d = 3, this result was proved in [3]. The three-dimensional result and its present extension build upon the method devised by [1].…”

“…Beck [1] found a specific estimate in this case, an estimate that is extended in [3]. In this note, the main technical device is the extension of this estimate, in the simplest instance, to arbitrary dimensions, see Lemma 5.2.…”

“…(Here, the average is in reference to the two functions we form the product of.) This lemma, in dimension d = 3 appears in [3]. We will give an inductive proof of this estimate, that requires that we revisit the three-dimensional case.…”

“…B(d − 1), d 4, we prove C(d). In this section, we will prove the estimates for C (3). As well, we present the inductive proof of C(d) assuming B(d − 1), for d 4. where the supremum is formed over all choices of M ⊂ H n 1 × H n 2 and all r functions subject to these conditions:…”

“…In dimension d = 3, [1,3] give partial information about this conjecture. In this paper, we can prove the following result, which appears to be new in dimensions d 4.…”