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Pisot and Salem numbers. (English) Zbl 0772.11041
Basel: Birkhäuser Verlag. ix, 291 p. (1992).
A real algebraic integer \(\alpha>1\) is called a Pisot number if all its remaining conjugates lie inside the unit circle, and a Salem number if it is not a Pisot number and its remaining conjugates lie inside or of the boundary of the unit circle. This monograph surveys the properties, applications and various generalizations of these two classes of numbers.
The first four chapters bring some material dealing with various rationality criteria for power series, Pisot’s theorem on compact families of rational functions, Schur’s algorithm (which gives conditions for a complex power series to represent a function holomorphic inside the unit circle and bounded there by unity) and uniform distribution \(\bmod 1\). One finds here some new results and proofs, e.g. a new criterion for hyper-rationality of complex power series (Theorem 3.1.1) and a new proof of Smyth’s theorem on the minimal value of the product of roots of a monic and non-reciprocal \(\mathbb{Z}\)-polynomial lying outside the unit circle.
Chapter 4 brings the definitions and main properties of the sets of Pisot and Salem numbers and in the next three chapters the distribution of these sets on the real line (limit points and small elements) as well as certain applications to some questions of uniform distribution and rational approximations are studied. Then various generalizations arrive: first on the real line (sets \(S_ q\) of Pisot) and then in the ring of adeles and in the rings of formal power series over an arbitrary field. A sequence \(a_ 0,a_ 1,\dots,\) of rational integers is called a Pisot sequence, if \(| a_{n+1}-a_ n^ 2 /a_{n+1}|<1/2\) holds. These sequences, the related Boyd sequences and their generalizations are considered in chapters 13 and 14.
The final chapter is devoted to applications of Pisot numbers to Fourier analysis and contains a proof of the Salem-Zygmund theorem: For \(\theta>2\) let \(E_ \theta=\{ \sum_{k\geq 0} \varepsilon_ k\theta^{-k}\): \(\varepsilon_ k=0,1\}\). The set \(E_ \theta\) is a set of uniqueness for trigonometric series if and only if \(\theta\) is a Pisot number.
The book is clearly written and is accessible also to non-specialists. The reviewer has only few rather minor critical remarks: the book has no indexes, which makes the reading sometimes uncomfortable, some notation remains unexplained (e.g. \(D(0,1)\) for the interior of the unit circle) and in the bibliography the abbreviations of some journals are distinct in various chapters (e.g. Matematičeskij Sbornik is called Math. Sb. in chapter 5 and simply Sbornik in chapter 15). The rather high price (158 sFr.) should also be mentioned here.

MSC:
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11B37 Recurrences
11K06 General theory of distribution modulo \(1\)
11R56 Adèle rings and groups
13F25 Formal power series rings
30B10 Power series (including lacunary series) in one complex variable
42A63 Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
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