Roth’s Orthogonal Function Method in Discrepancy Theory and Some New Connections Dmitriy Bilyk Abstract In this survey we give a comprehensive, but gentle introduction to the cir- cle of questions surrounding the classical problems of discrepancy theory, unified by the same approach originated in the work of Klaus Roth [85] and based on mul- tiparameter Haar (or other orthogonal) function expansions. Traditionally, the most important estimates of the discrepancy function were obtained using variations of this method. However, despite a large amount of work in this direction, the most im- portant questions in the subject remain wide open, even at the level of conjectures. The area, as well as the method, has enjoyed an outburst of activity due to the recent breakthrough improvement of the higher-dimensional discrepancy bounds and the revealed important connections between this subject and harmonic analysis, proba- bility (small deviation of the Brownian motion), and approximation theory (metric entropy of spaces with mixed smoothness). Without assuming any prior knowledge of the subject, we present the history and different manifestations of the method, its applications to related problems in various fields, and a detailed and intuitive outline of the latest higher-dimensional discrepancy estimate. 1 Introduction The subject and the structure of the present chapter is slightly unconventional. In- stead of building the exposition around the results from one area, united by a com- mon topic, we concentrate on problems from different fields which all share a com- mon method. The starting point of our discussion is one of the earliest results in discrepancy theory, Roth’s 1954 L2 bound of the discrepancy function in dimensions d ≥ 2 [85], as well as the tool employed to obtain this bound, which later evolved into a pow- Dmitriy Bilyk Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA e-mail: [email protected] 1 2 Dmitriy Bilyk erful orthogonal function method in discrepancy theory. We provide an extensive overview of numerous results in the subject of irregularities of distribution, whose proofs are based on this method, from the creation of the field to the latest achieve- ments. In order to highlight the universality of the method, we shall bring out and em- phasize analogies and connections of discrepancy theory and Roth’s method to prob- lems in a number of different fields, which include numerical integration (errors of cubature formulas), harmonic analysis (the small ball inequality), probability (small deviations of multiparameter Gaussian processes), approximation theory (metric en- tropy of spaces with dominating mixed smoothness). While some of these problems are related by direct implications, others are linked only by the method of proof, and perhaps new relations are yet to be discovered. We also present a very detailed and perceptive account of the proof of one of the most recent important developments in the theory, the improved L¥ bounds of the discrepancy function, and the corresponding improvements in other areas. We focus on the heuristics and the general strategy of the proof, and thoroughly ex- plain the idea of every step of this involved argument, while skipping some of the technicalities, which could have almost doubled the size of this chapter. We hope that the content of the volume will be of interest to experts and novices alike and will reveal the omnipotence of Roth’s method and the fascinating rela- tions between discrepancy theory and other areas of mathematics. We have made every effort to make our exposition clear, intuitive, and essentially self-contained, requiring familiarity only with the most basic concepts of the underlying fields. 1.1 The history and development of the field Geometric discrepancy theory seeks answers to various forms of the following ques- tions: How accurately can one approximate a uniform distribution by a finite dis- crete set of points? And what are the errors and limitations that necessarily arise in such approximations? The subject naturally grew out of the notion of uniform ¥ distribution in number theory. A sequence w = fwngn=1 ⊂ [0;1] is called uniformly distributed if, for any subinterval I ⊂ [0;1], the proportion of points wn that fall into I approximates its length, i.e. #fw 2 I : 1 ≤ n ≤ Ng lim n = jIj: (1) N!¥ N This property can be easily quantified using the notion of discrepancy: DN(w) = sup #fwn 2 I : 1 ≤ n ≤ Ng − N · jIj ; (2) I⊂[0;1] where I is an interval. In fact, it is not hard to show that w is uniformly distributed if and only if DN(w)=N tends to zero as N ! ¥ (see e.g. [64]). Roth’s Orthogonal Function Method 3 In [38, 1935], van der Corput posed a question whether there exists a sequence w for which the quantity DN(w) stays bounded as N gets large. More precisely, he mildly conjectured that the answer is “No” by stating that he is unaware of such sequences. Indeed, in [1, 1945], [2], van Aardenne-Ehrenfest gave a negative an- swer to this question, which meant that no sequence can be distributed too well. This result is widely regarded as a predecessor of the theory of irregularities of distribution. This area was turned into a real theory with precise quantitative estimates and conjectures by Roth, who in particular, see [85], greatly improved van Aardenne- Ehrenfest’s result by demonstrating that for any sequence w the inequality p DN(w) ≥ C logN (3) holds for infinitely many values of N. These results signified the birth of a new theory. Roth in fact worked on the following, more geometrical version of the problem. d Let PN ⊂ [0;1] be a set of N points and consider the discrepancy function DN(x1;:::;xd) = #fPN \ [0;x1) × ··· × [0;xd)g − N · x1 ····· xd; (4) i.e. the difference of the actual and expected number of points of PN in the box [0;x1) × ··· × [0;xd). Notice that, in contrast to some of the standard references, we are working with the unnormalized version of the discrepancy function, i.e. we do not divide this difference by N as it is often done. Obviously, the most natural norm ¥ d of this function is the L norm, i.e. the supremum of jDN(x)jover x 2 [0;1] , often referred to as the star-discrepancy. In fact the term star-discrepancy is reserved for 1 the sup-norm of the normalized discrepancy function, i.e. N kDNk¥, however since we only use the unnormalized version in this text, we shall abuse the language and apply this term to kDNk¥. ¥ Instead of directly estimating the L norm of the discrepancy function kDNk¥ = 2 sup DN(x) , Roth considered a smaller quantity, namely its L norm kDNk2 . This x2[0;1]d substitution allowed for an introduction of a variety of Hilbert space techniques, including orthogonal decompositions. In this setting Roth proved Theorem 1 (Roth, 1954, [85]). In all dimensions d ≥ 2, for any N-point set PN ⊂ [0;1]d, one has d−1 2 DN 2 ≥ Cd log N; (5) where Cd is an absolute constant that depends only on the dimension d. This in particular implies that d−1 sup DN(x) ≥ Cd log 2 N: (6) x2[0;1]d It was also shown that, when d = 2, inequality (6) is equivalent to (3). More gen- erally, uniform lower bounds for the discrepancy function of finite point sets (for 4 Dmitriy Bilyk all values of N) in dimension d are equivalent to lower estimates for the discrep- ancy of infinite sequences (2) (for infinitely many values of N) in dimension d − 1. These two settings are sometimes referred to as ‘static’ (fixed finite point sets) and ’dynamic’ (infinite sequences). In these terms, one can say that the dynamic and static problems are equivalent at the cost of one dimension – the relation becomes intuitively clear if one views the index of the sequence (or time) as an additional dimension. In this text, we adopt the former geometrical, ‘static’ formulation of the problems. According to Roth’s own words, these results “started a new theory” [34]. The paper [85] in which it was presented, entitled “On irregularities of distribution”, has had a tremendous influence on the further development of the field. Even the number of papers with identical or similar titles, that appeared in subsequent years, attests to its importance: 4 papers by Roth himself (On irregularities of distribution. I-IV, [85, 86, 87, 88]), one by H. Davenport (Note on irregularities of distribution, [42]), 10 by W. M. Schmidt (Irregularities of distribution. I-X, [90, 91, 92, 93, 94, 95, 96, 97, 98, 99]), 2 by J. Beck (Note on irregularities of distribution. I-II, [6, 7]), 4 by W. W. L. Chen (On irregularities of distribution. I-IV, [29, 30, 31, 32]), at least 2 by Beck and Chen (Note on irregularities of distribution. I-II, [10, 12] and several others with similar, but more specific names, as well as the fundamental monograph on the subject by Beck and Chen, “Irregularities of distribution”, [11]. The technique proposed in the aforementioned paper was no less important than the results themselves. Roth was the first to apply the expansion of the discrepancy function DN in the classical orthogonal Haar basis. Furthermore, he realized that in order to obtain good estimates of kDNk2 it suffices to consider just its projection onto the span of those Haar functions which are supported on dyadic rectangles of 1 volume roughly equal to N . This is heuristically justified by the fact that, for a well distributed set, each such rectangle contains approximately one point. To be even 1 more precise, the size of the rectangles R was chosen so that jRj ≈ 2N , ensuring that about half of all rectangles are free of points of PN.