##
**Irregularities of distribution.**
*(English)*
Zbl 0617.10039

Cambridge Tracts in Mathematics, 89. Cambridge etc.: Cambridge University Press. XIV, 294 p.; £32.50; $ 54.50 (1987).

This monograph is devoted to the theory of irregularities of point distributions in the \(K\)-dimensional Euclidean unit cube \(U^K\) and several generalizations. The book is dedicated to K. F. Roth and W. M. Schmidt, who have heavily influenced this branch of mathematics.

Starting point of the theory was a conjecture of Van der Corput (1935): Let \((s_n)\) be a sequence of points in \(U^1\), and

\[ D(N,s_n)=\sup_{0\leq x<1}| \text{card}\{1\leq n\leq N: 0\leq s_ n<x\}-Nx| \]

denotes the discrepancy of \((s_n)\); then \(\limsup_{N\to \infty}D(N,s_n)=\infty\).

In short the conjecture means that a sequence cannot be too evenly distributed. A first proof was given by T. Van Aardenne-Ehrenfest (1945). K. F. Roth (1954) was able to prove a refinement of this conjecture by means of his “orthogonal function method”. Let \(\mathfrak P\) be a point distribution in \(U^K\) and \(Z=Z(\mathfrak P,B(x_1\dots x_K))\) the numbers of points in the box \[ B(x_1 \dots x_K)=\{(y_1,\dots,y_K): 0\leq y_j<x_j,\;j=1,\dots,K\}. \]

Then for \(0<W\leq \infty\) one may consider the \(L^\)-norm (“\(L^W\)-discrepancy”)

\[ \| D(\mathfrak P)\|_W= \left(\int_{U^K} | Z-Nx_1 \dots x_K|^W\,dx_1 \dots dx_K\right)^{1/W}. \]

Roth established the estimates \(D(K,2,N)\gg (\log N)^{(K-1)/2}\) and \(D(K,\infty,N)\gg (\log N)^{(K- 1)/2}\), where \(D(K,W,N)\) denotes the infimum of \(D(\mathfrak P)\) over all \(K\)- dimensional point distributions consisting of \(N\) points. In the one-dimensional case it immediately follows that \(D(N,s_n)\gg \sqrt{\log N}\) for infinitely many \(N\).

In Chapter 2 W. M. Schmidt’s modification of Roth’s method and also Halász’ variation of Roth’s method are discussed. The main results are lower bounds for \(D(K,W,N)\), \(W\geq 1\). Furthermore Halász’ method is applied to a colouring problem in \(U^2\) (Beck 1981, Roth 1985).

In Chapter 3 some upper bounds are established: Davenport’s bound \(D(2,2,N)\ll \sqrt{\log N}\) and Roth’s bound \(D(3,2,N)\ll \log N\) as well as the second author’s general bound

\[ D(K,W,N)\ll (\log N)^{(K-1)/2}\quad (1<W<\infty), \] which proves that Schmidt’s lower bound for \(D(K,W,N)\) is best possible. However, in the case \(W=\infty\), \(K>2\) it is conjectured that

\[ D(K,W,N)\gg (\log N)^{K-1}\quad \text{(``great\;open\;problem'')}. \]

Because of Halton’s construction this estimate would be best possible. The proofs are either based on diophantine approximation or on probabilistic methods.

Chapter 4 is devoted to W. M. Schmidt’s final answer to the Van der Corput conjecture. By a combinatorial method he obtained the best possible estimate \(D(N,s_n)\gg \log N\) for infinitely many \(N\). Several further applications of this method are given as well as G. Wagner’s solution of a problem of Erdős in diophantine approximation.

From Chapter 5 onwards several generalizations of these classical problems are discussed: e.g. the points in \(U^K\) are no longer counted in rectangular boxes but in circular discs or in convex bodies, or the points are taken from the unit ball or from the unit sphere. To attack such problems there exist two general methods: W. M. Schmidt’s “integral equation method” and the first author’s recent Fourier transform approach. The “integral equation method”, for instance, works to establish lower bounds for the discrepancy in \(U^K\) with respect to balls or lower bounds for the discrepancy on the sphere with respect to spherical caps; see W. M. Schmidt [Lectures on irregularities of distribution. Bombay: Tata Inst. Fundamental Research (1977; Zbl 0434.10031)]. The so called Fourier transform method was developed by the first author [Invent. Math. 74, 477–487 (1983; Zbl 0528.10037)]. It works in many cases, e.g. for lower bounds in the unit ball with respect to spherical caps. This method makes heavily use of harmonic analysis techniques: Gauss kernel, Fejér kernel etc. Especially in the two-dimensional case several discrepancy bounds are obtained by applying known facts from discrete geometry (approximability number, geometric inequalities).

In the final chapter Roth’s famous “1/4-theorem” on irregularities of integer sequences relative to arithmetic progressions is shown. The book is concluded with a list of some open problems.

The obvious merit of the book is the fact that the ideas behind the technical calculations of the proofs are worked out. The authors wrote a clear and fascinating monograph on the theory of irregularities of distribution: useful for experts as a stimulation to further research and also for beginners to get familiar with this nice branch of mathematics.

Starting point of the theory was a conjecture of Van der Corput (1935): Let \((s_n)\) be a sequence of points in \(U^1\), and

\[ D(N,s_n)=\sup_{0\leq x<1}| \text{card}\{1\leq n\leq N: 0\leq s_ n<x\}-Nx| \]

denotes the discrepancy of \((s_n)\); then \(\limsup_{N\to \infty}D(N,s_n)=\infty\).

In short the conjecture means that a sequence cannot be too evenly distributed. A first proof was given by T. Van Aardenne-Ehrenfest (1945). K. F. Roth (1954) was able to prove a refinement of this conjecture by means of his “orthogonal function method”. Let \(\mathfrak P\) be a point distribution in \(U^K\) and \(Z=Z(\mathfrak P,B(x_1\dots x_K))\) the numbers of points in the box \[ B(x_1 \dots x_K)=\{(y_1,\dots,y_K): 0\leq y_j<x_j,\;j=1,\dots,K\}. \]

Then for \(0<W\leq \infty\) one may consider the \(L^\)-norm (“\(L^W\)-discrepancy”)

\[ \| D(\mathfrak P)\|_W= \left(\int_{U^K} | Z-Nx_1 \dots x_K|^W\,dx_1 \dots dx_K\right)^{1/W}. \]

Roth established the estimates \(D(K,2,N)\gg (\log N)^{(K-1)/2}\) and \(D(K,\infty,N)\gg (\log N)^{(K- 1)/2}\), where \(D(K,W,N)\) denotes the infimum of \(D(\mathfrak P)\) over all \(K\)- dimensional point distributions consisting of \(N\) points. In the one-dimensional case it immediately follows that \(D(N,s_n)\gg \sqrt{\log N}\) for infinitely many \(N\).

In Chapter 2 W. M. Schmidt’s modification of Roth’s method and also Halász’ variation of Roth’s method are discussed. The main results are lower bounds for \(D(K,W,N)\), \(W\geq 1\). Furthermore Halász’ method is applied to a colouring problem in \(U^2\) (Beck 1981, Roth 1985).

In Chapter 3 some upper bounds are established: Davenport’s bound \(D(2,2,N)\ll \sqrt{\log N}\) and Roth’s bound \(D(3,2,N)\ll \log N\) as well as the second author’s general bound

\[ D(K,W,N)\ll (\log N)^{(K-1)/2}\quad (1<W<\infty), \] which proves that Schmidt’s lower bound for \(D(K,W,N)\) is best possible. However, in the case \(W=\infty\), \(K>2\) it is conjectured that

\[ D(K,W,N)\gg (\log N)^{K-1}\quad \text{(``great\;open\;problem'')}. \]

Because of Halton’s construction this estimate would be best possible. The proofs are either based on diophantine approximation or on probabilistic methods.

Chapter 4 is devoted to W. M. Schmidt’s final answer to the Van der Corput conjecture. By a combinatorial method he obtained the best possible estimate \(D(N,s_n)\gg \log N\) for infinitely many \(N\). Several further applications of this method are given as well as G. Wagner’s solution of a problem of Erdős in diophantine approximation.

From Chapter 5 onwards several generalizations of these classical problems are discussed: e.g. the points in \(U^K\) are no longer counted in rectangular boxes but in circular discs or in convex bodies, or the points are taken from the unit ball or from the unit sphere. To attack such problems there exist two general methods: W. M. Schmidt’s “integral equation method” and the first author’s recent Fourier transform approach. The “integral equation method”, for instance, works to establish lower bounds for the discrepancy in \(U^K\) with respect to balls or lower bounds for the discrepancy on the sphere with respect to spherical caps; see W. M. Schmidt [Lectures on irregularities of distribution. Bombay: Tata Inst. Fundamental Research (1977; Zbl 0434.10031)]. The so called Fourier transform method was developed by the first author [Invent. Math. 74, 477–487 (1983; Zbl 0528.10037)]. It works in many cases, e.g. for lower bounds in the unit ball with respect to spherical caps. This method makes heavily use of harmonic analysis techniques: Gauss kernel, Fejér kernel etc. Especially in the two-dimensional case several discrepancy bounds are obtained by applying known facts from discrete geometry (approximability number, geometric inequalities).

In the final chapter Roth’s famous “1/4-theorem” on irregularities of integer sequences relative to arithmetic progressions is shown. The book is concluded with a list of some open problems.

The obvious merit of the book is the fact that the ideas behind the technical calculations of the proofs are worked out. The authors wrote a clear and fascinating monograph on the theory of irregularities of distribution: useful for experts as a stimulation to further research and also for beginners to get familiar with this nice branch of mathematics.

Reviewer: Robert F. Tichy (Graz)

### MSC:

11K38 | Irregularities of distribution, discrepancy |

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

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