Sets of multiples.

*(English)*Zbl 0871.11001
Cambridge Tracts in Mathematics. 118. Cambridge: Cambridge University Press. xvi, 264 p. (1996).

There have been many developments in the theory of sets of multiples since the publication of [Sequences, Oxf., Clarendon Press (1966; Zbl 0141.04405)] by H. Halberstam and K. F. Roth, and this book seeks to present some of them. Its underlying theme is an investigation of properties of sequences of integers that are multiples of an element in a given base sequence. Thus if \(\mathcal A\) is a non-decreasing sequence of positive integers, and \(\tau(n,{\mathcal A})=\text{card}\{a:a|n, a\in {\mathcal A}\}\), the author studies the set
\[
M({\mathcal A})=\{n:n\in\mathbb{Z}^+, \tau(n,{\mathcal A})\geq 1\},
\]
regarding it as a non-decreasing sequence. This sequence has asymptotic density \(dM({\mathcal A})\) if
\[
\text{card}\{n\leq x: n\in M({\mathcal A})\}\sim dM({\mathcal A})x\quad\text{as }x\to\infty,
\]
and logarithmic density \(\delta M({\mathcal A})\) if
\[
\sum_{n\leq x,\;n\in M({\mathcal A})} n^{-1} \sim \delta M({\mathcal A}) \log x \quad\text{as }x\to\infty.
\]
The base sequence \(\mathcal A\) is called a Besicovitch sequence if \(dM({\mathcal A})\) exists, and, when \(1\notin {\mathcal A}\), a Behrend sequence if \(\delta M({\mathcal A})=1\). These definitions provide the starting point for much of the material in this book.

The study of sets of multiples can be regarded as having its roots in investigations into abundant numbers begun over 60 years ago. From then onwards, the influence of the work of Paul Erdös has been very apparent. In more recent times, apart from the contribution by Erdös, many other key results established in this book are due to its author and/or Gérald Tenenbaum. Indeed this volume is a companion to [Divisors, Cambridge Tracts Math. 90, Cambridge Univ. Press (1988; Zbl 0653.10001)] by R. R. Hall and G. Tenenbaum, and some of the concepts considered there also arise here.

The author gives a coherent account of the theory of sets of multiples from today’s perspective, going right up to deep current research and including, when appropriate, a formulation of various unsolved problems. The book is in a sense self-contained, but a reader with a good general background in the variety of techniques used, including elementary, analytic and probabilistic methods, would have a distinct advantage and would still find the material challenging. Care has been taken to provide explanations of the aims and background of each major development discussed.

Some classical results are established in chapter 0, and chapter 1 is concerned with deriving conditions under which a sequence \(\mathcal A\) is, or is not, Besicovitch or Behrend. The second chapter deals with derived sequences, where the \(k\)-th derived sequence \({\mathcal A}^{(k)}\) of \(\mathcal A\) consists of all possible lowest common multiples of \(k+1\) elements of \(\mathcal A\); thus \(n\in M({\mathcal A}^{(k)})\) if and only if \(\tau(n,{\mathcal A}) > k\). A quantity related to the discrepancy \[ E(x,{\mathcal A}) =\text{card}\{m\leq x: m\in M({\mathcal A})\}-dM({\mathcal A})x \] is investigated in chapter 3, with the cases when the primitive sequence associated with \(\mathcal A\) is finite or infinite treated separately. The approach in these two chapters is more combinatorial. Chapter 4 is entitled probabilistic group theory, and culminates in §4.4 in the construction of Behrend sequences depending on \(|\log a|\) and the value of \(\Omega(a)\), the number of prime factors of \(a\) (with multiplicity). Chapters 5, 6, 7 are more analytical in nature. In chapter 5, conditions are established under which a sequence \(\mathcal A\) has divisor density \(D{\mathcal A}\), that is \[ \tau(n,{\mathcal A})=(D{\mathcal A}+o(1))\tau(n)\quad\text{p.p.} \] as \(n\to\infty\), where \(\tau(n)=\tau(n,\mathbb{Z}^+)\). Let \[ {\mathcal A}(z)={\mathcal A}(z,f)=\{a:0 < f(a)\leq z \pmod 1\},\quad\text{where }f;\mathbb{Z}^+\to\mathbb{R}, \] and put \[ \Delta(n,f)=\sup_{0<w<z\leq 1}|\tau(n,{\mathcal A}(z))-\tau(n,{\mathcal A}(w))-(z-w)\tau(n)|. \] Then \(f\) is divisor uniformly distributed if for every \(\varepsilon>0\) \[ \Delta(n,f)<\varepsilon\tau(n)\quad\text{p.p.} \] Examples of such functions are given in chapter 6, and criteria under which \(f\) is divisor uniformly distributed found. In chapter 7, a conjecture made in §2.2 of [Divisors, loc. cit.] concerning the function \[ H(x,y,z)=\text{card}\{n\leq x:\exists\;d|n\text{ with }y < d \leq z\} \] is settled.

The chapter contents outlined above indicate the wide scope of the topics covered and the different methods of proof utilized. Research workers and graduate students working in number theory and related areas will find this text an invaluable source and one likely to remain an important reference book for some considerable time. It is well written, especially considering the very technical nature of the content, although inevitably it is very demanding on the reader. Altogether this volume is a valuable and welcome addition to the literature in the field.

The study of sets of multiples can be regarded as having its roots in investigations into abundant numbers begun over 60 years ago. From then onwards, the influence of the work of Paul Erdös has been very apparent. In more recent times, apart from the contribution by Erdös, many other key results established in this book are due to its author and/or Gérald Tenenbaum. Indeed this volume is a companion to [Divisors, Cambridge Tracts Math. 90, Cambridge Univ. Press (1988; Zbl 0653.10001)] by R. R. Hall and G. Tenenbaum, and some of the concepts considered there also arise here.

The author gives a coherent account of the theory of sets of multiples from today’s perspective, going right up to deep current research and including, when appropriate, a formulation of various unsolved problems. The book is in a sense self-contained, but a reader with a good general background in the variety of techniques used, including elementary, analytic and probabilistic methods, would have a distinct advantage and would still find the material challenging. Care has been taken to provide explanations of the aims and background of each major development discussed.

Some classical results are established in chapter 0, and chapter 1 is concerned with deriving conditions under which a sequence \(\mathcal A\) is, or is not, Besicovitch or Behrend. The second chapter deals with derived sequences, where the \(k\)-th derived sequence \({\mathcal A}^{(k)}\) of \(\mathcal A\) consists of all possible lowest common multiples of \(k+1\) elements of \(\mathcal A\); thus \(n\in M({\mathcal A}^{(k)})\) if and only if \(\tau(n,{\mathcal A}) > k\). A quantity related to the discrepancy \[ E(x,{\mathcal A}) =\text{card}\{m\leq x: m\in M({\mathcal A})\}-dM({\mathcal A})x \] is investigated in chapter 3, with the cases when the primitive sequence associated with \(\mathcal A\) is finite or infinite treated separately. The approach in these two chapters is more combinatorial. Chapter 4 is entitled probabilistic group theory, and culminates in §4.4 in the construction of Behrend sequences depending on \(|\log a|\) and the value of \(\Omega(a)\), the number of prime factors of \(a\) (with multiplicity). Chapters 5, 6, 7 are more analytical in nature. In chapter 5, conditions are established under which a sequence \(\mathcal A\) has divisor density \(D{\mathcal A}\), that is \[ \tau(n,{\mathcal A})=(D{\mathcal A}+o(1))\tau(n)\quad\text{p.p.} \] as \(n\to\infty\), where \(\tau(n)=\tau(n,\mathbb{Z}^+)\). Let \[ {\mathcal A}(z)={\mathcal A}(z,f)=\{a:0 < f(a)\leq z \pmod 1\},\quad\text{where }f;\mathbb{Z}^+\to\mathbb{R}, \] and put \[ \Delta(n,f)=\sup_{0<w<z\leq 1}|\tau(n,{\mathcal A}(z))-\tau(n,{\mathcal A}(w))-(z-w)\tau(n)|. \] Then \(f\) is divisor uniformly distributed if for every \(\varepsilon>0\) \[ \Delta(n,f)<\varepsilon\tau(n)\quad\text{p.p.} \] Examples of such functions are given in chapter 6, and criteria under which \(f\) is divisor uniformly distributed found. In chapter 7, a conjecture made in §2.2 of [Divisors, loc. cit.] concerning the function \[ H(x,y,z)=\text{card}\{n\leq x:\exists\;d|n\text{ with }y < d \leq z\} \] is settled.

The chapter contents outlined above indicate the wide scope of the topics covered and the different methods of proof utilized. Research workers and graduate students working in number theory and related areas will find this text an invaluable source and one likely to remain an important reference book for some considerable time. It is well written, especially considering the very technical nature of the content, although inevitably it is very demanding on the reader. Altogether this volume is a valuable and welcome addition to the literature in the field.

Reviewer: E.J.Scourfield (Egham)

##### MSC:

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

11B83 | Special sequences and polynomials |

11N25 | Distribution of integers with specified multiplicative constraints |

11B05 | Density, gaps, topology |