Let h R denote an L ∞ normalized Haar function adapted to a dyadic rectangle R ⊂ [0, 1] d . We show that for choices of coefficients α(R), we have the following lower bound on the L ∞ norms of the sums of such functions, where the sum is over rectangles of a fixed volume:The point of interest is the dependence upon the logarithm of the volume of the rectangles. With n (d−1)/2 on the left above, the inequality is trivial, while it is conjectured that the inequality holds with n (d−2)/2 . This is known in the case of d = 2 [Michel Talagrand, The small ball problem for the Brownian sheet, Ann. Probab. 22 (3) (1994) 1331-1354, MR 95k:60049], and a recent paper of two of the authors [Dmitriy Bilyk, Michael T. Lacey, On the Small Ball Inequality in three dimensions, Duke Math. J., (2006), in press, arXiv: math.CA/0609815] proves a partial result towards the conjecture in three dimensions. In this paper, we show that the argument of [Dmitriy Bilyk, Michael T. Lacey, On the Small Ball Inequality in three dimensions, Duke Math. J. , (2006), in press, arXiv: math.CA/0609815] can be extended to arbitrary dimension. We also prove related results in the subjects of the irregularity of distribution, and approximation theory. The authors are unaware of any prior results on these questions in any dimension d 4.
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...
The classical Stolarsky invariance principle connects the spherical cap L 2 discrepancy of a finite point set on the sphere to the pairwise sum of Euclidean distances between the points. In this paper we further explore and extend this phenomenon. In addition to a new elementary proof of this fact, we establish several new analogs, which relate various notions of discrepancy to different discrete energies. In particular, we find that the hemisphere discrepancy is related to the sum of geodesic distances. We also extend these results to arbitrary measures on the sphere and arbitrary notions of discrepancy and apply them to problems of energy optimization and combinatorial geometry and find that, surprisingly, the geodesic distance energy behaves differently than its Euclidean counterpart.
Abstract. Let A N be an N-point set in the unit square and consider the Discrepancy function, and |[ 0, x)| denotes the Lebesgue measure of the rectangle. We give various refinements of a well-known result of (Schmidt, 1972) The case of α = ∞ is the Theorem of Schmidt. This estimate is sharp. For the digit-scrambled van der Corput sequence, we havewhenever N = 2 n for some positive integer n. This estimate depends upon variants of the Chang-Wilson-Wolff inequality (Chang et al., 1985). We also provide similar estimates for the BMO norm of D N .
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