##
**Sequences, discrepancies and applications.**
*(English)*
Zbl 0877.11043

Lecture Notes in Mathematics. 1651. Berlin: Springer. xiii, 503 p. (1997).

The theory of uniform distribution of sequences was surveyed in the 1970s in the books of L. Kuipers and H. Niederreiter [Uniform distribution of sequences, Wiley, New York (1974; Zbl 0281.10001)] and E. Hlawka [Theorie der Gleichverteilung, Mannheim, Bibliographisches Institut (1979; Zbl 0406.10001)]. Since that time this theory has undergone a vigorous development and it has generated a lot of attention because of its many applications to numerical analysis, scientific computing, and other fields. Some of these applications were summarized a few years ago in the lecture notes of the reviewer [Random number generation and quasi-Monte Carlo methods, SIAM, Philadelphia (1992; Zbl 0761.65002)].

The monograph under review is the outcome of an impressive effort to update the books above and to present the state-of-the-art in the theory of uniform distribution of sequences and its applications. The three chapters (Discrepancy of sequences, General concepts of uniform distribution, Applications) cover practically every aspect of the subject. The main text stresses results that have been obtained in the last 20 years, but in the accompanying bibliographical notes the connection with earlier work is made, so that a full picture of all relevant developments emerges.

Chapter 1 discusses the quantitative theory of uniform distribution of sequences. Basic concepts such as discrepancy and dispersion are introduced and the essential tools of the trade (exponential sums, Erdös-Turán inequality) are reviewed. There is a detailed coverage of irregularities of distribution and of discrepancy bounds for special sequences. Considerable space is also given to probabilistic results, such as the law of the iterated logarithm for the discrepancy of lacunary sequences. Chapter 2 is devoted to geometric and abstract aspects of the theory and to extensions of the notion of uniform distribution to summation methods and to continuous uniform distribution of functions. A particularly nice section deals with uniform distribution of sequences in discrete spaces, a subject with interesting combinatorial ramifications. Chapter 3 presents various applications to numerical analysis, spherical designs, pseudorandom number generation, and real-world problems (e.g. from mathematical finance). The classical application of uniformly distributed sequences to numerical integration is of course a central topic here, but lattice rules – an important development in the last ten years – receive only a cursory treatment. The book includes a monumental bibliography which stretches over 65 pages and is a real treasure trove.

It is inevitable in such a big work that errors have sneaked in. On page 131 the authors omit the fact that the dispersion spectrum was introduced and analyzed in the reviewer’s paper [Topics in classical number theory, Colloq. Budapest 1981, Vol. II, Colloq. Math. Soc. Janos Bolyai 34, 1163-1208 (1984; Zbl 0547.10045)]. The famous lecture of Erdös was held at Nijenrode and not at Nijmrode (see page 185). On page 386 the bound \(T_q(s) =O(s)\) holds for any base \(q\) and not just for \(q=2\), and the reference [1368] has to be replaced by [1370] and [1317] by [1319]. On page 398 the reference [1369] has to be replaced by [1370]. The names Devroye, Eichenauer-Herrmann, and Patarin are misspelled on page 432. In the bibliography there are errors concerning authors (spelling errors, missing initials, wrong initials, etc.) in the entries [196], [452], [522]–[555] (a particularly irritating systematic error), [626], [1269], [1354], [1870], [1871], and [1947]. The references [1961] and [1967] are identical. However, these quibbles should not detract from the immense value of the book as an indispensable source of information on uniformly distributed sequences and their applications.

The monograph under review is the outcome of an impressive effort to update the books above and to present the state-of-the-art in the theory of uniform distribution of sequences and its applications. The three chapters (Discrepancy of sequences, General concepts of uniform distribution, Applications) cover practically every aspect of the subject. The main text stresses results that have been obtained in the last 20 years, but in the accompanying bibliographical notes the connection with earlier work is made, so that a full picture of all relevant developments emerges.

Chapter 1 discusses the quantitative theory of uniform distribution of sequences. Basic concepts such as discrepancy and dispersion are introduced and the essential tools of the trade (exponential sums, Erdös-Turán inequality) are reviewed. There is a detailed coverage of irregularities of distribution and of discrepancy bounds for special sequences. Considerable space is also given to probabilistic results, such as the law of the iterated logarithm for the discrepancy of lacunary sequences. Chapter 2 is devoted to geometric and abstract aspects of the theory and to extensions of the notion of uniform distribution to summation methods and to continuous uniform distribution of functions. A particularly nice section deals with uniform distribution of sequences in discrete spaces, a subject with interesting combinatorial ramifications. Chapter 3 presents various applications to numerical analysis, spherical designs, pseudorandom number generation, and real-world problems (e.g. from mathematical finance). The classical application of uniformly distributed sequences to numerical integration is of course a central topic here, but lattice rules – an important development in the last ten years – receive only a cursory treatment. The book includes a monumental bibliography which stretches over 65 pages and is a real treasure trove.

It is inevitable in such a big work that errors have sneaked in. On page 131 the authors omit the fact that the dispersion spectrum was introduced and analyzed in the reviewer’s paper [Topics in classical number theory, Colloq. Budapest 1981, Vol. II, Colloq. Math. Soc. Janos Bolyai 34, 1163-1208 (1984; Zbl 0547.10045)]. The famous lecture of Erdös was held at Nijenrode and not at Nijmrode (see page 185). On page 386 the bound \(T_q(s) =O(s)\) holds for any base \(q\) and not just for \(q=2\), and the reference [1368] has to be replaced by [1370] and [1317] by [1319]. On page 398 the reference [1369] has to be replaced by [1370]. The names Devroye, Eichenauer-Herrmann, and Patarin are misspelled on page 432. In the bibliography there are errors concerning authors (spelling errors, missing initials, wrong initials, etc.) in the entries [196], [452], [522]–[555] (a particularly irritating systematic error), [626], [1269], [1354], [1870], [1871], and [1947]. The references [1961] and [1967] are identical. However, these quibbles should not detract from the immense value of the book as an indispensable source of information on uniformly distributed sequences and their applications.

Reviewer: H.Niederreiter (Wien)

### MSC:

11K06 | General theory of distribution modulo \(1\) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11K38 | Irregularities of distribution, discrepancy |

65C10 | Random number generation in numerical analysis |

65D30 | Numerical integration |

11K45 | Pseudo-random numbers; Monte Carlo methods |