Probabilistic analysis of multivariate problems

The development of the theory of uniform distribution modulo one and discrepancy the- ory started with Hermann Weyl`s seminal paper of 1916. Since then, it turned out that these notions are not only remarkably important concepts, but can be conveniently used for many problems of applied mathematics. For example, the so-called Quasi-Monte Carlo (QMC) method for numerical integration is based on the fact that the difference between the average value of a function, evaluated at certain sampling points, and its integral can be bounded by the variation of this function and the discrepancy of the set of sampling points. This observation, whose precise version is called the Koksma-Hlawka inequality, suggests that low-discrepancy point sets provide small integration errors in numerical integration. Since there exist constructions of point sets whose discrepancy is almost of asymptotic order N -1 , this means that QMC integration can perform significantly better than Monte Carlo (MC) integration (where the sampling points are randomly drawn, and the probabilistic error is of asymptotic order N -1/2). Although discrepancy theory is an intensively investigated field of mathematics, a num- ber of important problems are still open. In particular, many known results only provide reasonable error bounds if the number of sampling points is very large, and do not imply that QMC integration should work also for a moderate number of points (in compari- son with the dimension). This observation recently led to an increased interest in error bounds for QMC integration in high dimensions, when the number of sampling points cannot be chosen large enough for an application of the classical discrepancy bounds. While classical constructions of low-discrepancy sequences are usually of purely de- terministic nature, a good deal of probability theory is used in these new methods, either by randomizing classical constructions, or by proving the existence of appropri- ate points sets in a random environment. Our research project will focus on applications of probabilistic methods for high- dimensional problems, particularly in the context of discrepancy theory. In the first year we will investigate so-called lacunary sequences of functions, which by a classical heuristics imitate many properties of independent random variables. The behavior of such lacunary sequences has never been investigated in detail in a multidimensional setting, which is surprising, since it can be hoped that the similarity between such sys- tems and independent random variables extends to the multivariate case, and therefore lacunary sequences could serve as an alternative for MC integration. In the second year we will study the probabilistic properties of multivariate random sequences in detail, in particular the deviation between the empirical distribution function and the under- lying distribution. Finally, in the last year, we will mainly focus on discrepancy theory, and use probabilistic methods to prove error bounds for the discrepancy of randomized QMC sequences and to investigate the existence of low-discrepancy points sets with "few" elements in comparison with the dimension.

One of the deepest results of Fourier analysis is Carlesons theorem (1966), stating that the Fourier series of any square integrable function is convergent with the exception of a set with Lebesgue measure 0. In contrast, very little is known about the case when the trigonometric functions sin x, cos x are replaced by a general periodic function f (x). In particular, there exist no satisfactory criteria for the almost everywhere convergence of series ck f (kx) and in the case of divergence, sharp results for the speed of growth of its partial sums, even though these problems have been investigated intensively since the 1920s. Several well known open problems of analysis can be formulated in this eld and the most signicant results of our project are the solution of some of these problems. Among others, we found optimal or near optimal convergence criteria for ck f (kx) in the case when the function f has bounded variation or the Fourier coecients of f decrease regularly, with polynomial speed. N By a famous conjecture of Khinchin (1923), the averages N -1k=1 f (kx) converge to zero almost everywhere provided that f is periodic with zero Lebesgue integral on its period interval. This conjecture was disproved by Marstrand (1970), but until today, no characterization of functions f satisfying this convergence relation has been found. One main result of our project states that the well known sucient conditiona2k s-1 (k) < 8 of Koksma (1953) for the Fourier coecients of f is optimal, closing the near 100 year long history of the problem in the case f L2 . In the case when f is a centered indicator function (extended periodically), the last partial sums are closely related to the discrepancy of the sequence {nx}. This sequence is a classical object of mathematical analysis and starting with the works of Hardy and Littlewood, Khinchin, Weyl, Ostrowski and others in the early 1900s, its behavior has been completely cleared up. On the other hand, very little is known about the behavior of the discrepancy of subsequences {nk x} and as a consequence of our studies of the system f (nx) mentioned above, we got substantial information in this eld, too. The proofs of the previous results use tools from various parts of mathematics, such as number theory, complex analysis, ergodic theory, probability theory and as a byproduct of our results mentioned above, we got several results in these elds as well. Among others, we got optimal bounds for so called GCD (greatest common divisor) sums, closing a long series of investigations started in the 1930s; we proved limit theorems in the extremal theory of continued fractions and strong approximation theorems for Bernoulli shift processes. We also extended the classical Paley- Zygmund theory of random trigonometric series for the case of series with random gaps and proved results for irregularities of stochastic games.