the attention of The publication of Charles Pisot's thesis in 1938 brought to the mathematical community those marvelous numbers now known as the Pisot numbers (or the Pisot-Vijayaraghavan numbers). Although these numbers had been discovered earlier by A. Thue and then by G. H. Hardy, it was Pisot's result in that paper of 1938 that provided the link to harmonic analysis, as discovered by Raphael Salem and described in a series of papers in the 1940s. In one of these papers, Salem introduced the related class of numbers, now universally known as the Salem numbers. These two sets of algebraic…mehr
the attention of The publication of Charles Pisot's thesis in 1938 brought to the mathematical community those marvelous numbers now known as the Pisot numbers (or the Pisot-Vijayaraghavan numbers). Although these numbers had been discovered earlier by A. Thue and then by G. H. Hardy, it was Pisot's result in that paper of 1938 that provided the link to harmonic analysis, as discovered by Raphael Salem and described in a series of papers in the 1940s. In one of these papers, Salem introduced the related class of numbers, now universally known as the Salem numbers. These two sets of algebraic numbers are distinguished by some striking arith metic properties that account for their appearance in many diverse areas of mathematics: harmonic analysis, ergodic theory, dynamical systems and alge braic groups. Until now, the best known and most accessible introduction to these num bers has been the beautiful little monograph of Salem, Algebraic Numbers and Fourier Analysis, first published in 1963. Since the publication of Salem's book, however, there has been much progress in the study of these numbers. Pisot had long expressed the desire to publish an up-to-date account of this work, but his death in 1984 left this task unfulfilled.
1 Rational series.- 1.1 Algebraic criteria of rationality.- 1.2 Criteria of rationality in C.- 1.3 Generalized Fatou's lemma.- Notes.- References.- 2 Compact families of rational functions.- 2.1 Properties of formal series with rational coefficients.- 2.2 Compact families of rational functions.- Notes.- References.- 3 Meromorphic functions on D(0,1). Generalized Schur algorithm.- 3.0 Notation.- 3.1 Properties of Schur's determinants.- 3.2 Characterization of functions belonging to M.- 3.3 Generalized Schur algorithm.- 3.4 Characterization of certain meromorphic functions on D(0,1).- 3.5 Smyth's theorem.- Notes.- References.- 4 Generalities concerning distribution modulo 1 of real sequences.- 4.0 Notation and examples.- 4.1 Sequences with finitely many limit points modulo 1.- 4.2 Uniform distribution of sequences.- 4.3 Weyl's theorems.- 4.4 Van der Corput's and Fejer's theorems. Applications.- 4.5 Koksma's theorem.- 4.6 Some notions about uniform distribution modulo 1 in Rp.- Notes.- References.- 5 Pisot numbers, Salem numbers and distribution modulo 1.- 5.0 Notation.- 5.1 Some sequences (??n) non-uniformly distributed modulo 1.- 5.2 Pisot numbers and Salem numbers. Definitions and algebraic properties.- 5.3 Distribution modulo 1 of the sequences (?n) with ? a U-number.- 5.4 Pisot numbers and distribution modulo 1 of certain sequences (??n).- 5.5 Salem numbers and distribution modulo 1 of certain sequences (??n).- 5.6 Sequences (??n) non-uniformly distributed modulo 1.- Notes.- References.- 6 Limit points of Pisot and Salem sets.- 6.0 Notation.- 6.1 Closure of the set S.- 6.2 The derived set S? of S.- 6.3 Successive derived sets of S.- 6.4 Limit points of the set T.- Notes.- References.- 7 Small Pisot numbers.- 7.1 Schur's approximations forelements of N*1.- 7.2 Small Pisot numbers.- 7.3 The smallest number of S?.- Notes.- References.- 8 Some properties and applications of Pisot numbers.- 8.1 Some algebraic properties and applications of Pisot and Salem numbers.- 8.2 An application of Pisot numbers to a problem of uniform distribution.- 8.3 Application of Pisot numbers to a problem of rational approximations of algebraic numbers.- 8.4 Pisot numbers and the Jacobi-Perron algorithm.- Notes.- References.- 9 Algebraic number sets.- 9.1 Sq sets.- 9.2 n-tuples of algebraic numbers.- Notes.- References.- 10 Rational functions over rings of adeles.- 10.1 Adeles of Q.- 10.2 Analytic functions in Cp.- 10.3 Rationality criteria in QI[[X]].- 10.4 Compact families of rational functions.- Notes.- References.- 11 Generalizations of Pisot and Salem numbers to adeles.- 11.1 Definition of the set UI.- 11.2 Subsets of UI and characterizations.- 11.3 The sets SI?.- 11.4 The sets TI.- 11.5 The sets SIJ.- 11.6 The sets BI.- 11.7 Closed subsets of SI?.- 11.8 Limit points of the sets TI.- Notes.- References.- 12 Pisot elements in a field of formal power series.- 12.0 Generalities and notation.- 12.1 Definitions of the sets U and S.- 12.2 Characterizations of the sets U and S.- 12.3 Limit points of the sets U and S.- 12.4 Relation between the sets S and S.- Notes.- References.- 13 Pisot sequences, Boyd sequences and linear recurrence.- 13.0 Convergence theorems.- 13.1 Pisot sequences.- 13.2 Linear recurrence and Pisot sequences.- 13.3 Boyd sequences.- Notes.- References.- 14 Generalizations of Pisot and Boyd sequences.- 14.1 Convergence theorems in AI.- 14.2 Pisot sequences in AI.- 14.3 Boyd sequences in AI.- 14.4 Pisot and Boyd sequences in a field of formal power series.- Notes.- References.- l5 The Salem-Zygmundtheorem.- 15.1 Introduction.- 15.2 Sets of uniqueness.- 15.3 Symmetric perfect sets.- 15.4 The sufficient condition for the Salem-Zygmund theorem.- 15.5 A theorem by Senge and Strauss.- References.
1 Rational series.- 1.1 Algebraic criteria of rationality.- 1.2 Criteria of rationality in C.- 1.3 Generalized Fatou's lemma.- Notes.- References.- 2 Compact families of rational functions.- 2.1 Properties of formal series with rational coefficients.- 2.2 Compact families of rational functions.- Notes.- References.- 3 Meromorphic functions on D(0,1). Generalized Schur algorithm.- 3.0 Notation.- 3.1 Properties of Schur's determinants.- 3.2 Characterization of functions belonging to M.- 3.3 Generalized Schur algorithm.- 3.4 Characterization of certain meromorphic functions on D(0,1).- 3.5 Smyth's theorem.- Notes.- References.- 4 Generalities concerning distribution modulo 1 of real sequences.- 4.0 Notation and examples.- 4.1 Sequences with finitely many limit points modulo 1.- 4.2 Uniform distribution of sequences.- 4.3 Weyl's theorems.- 4.4 Van der Corput's and Fejer's theorems. Applications.- 4.5 Koksma's theorem.- 4.6 Some notions about uniform distribution modulo 1 in Rp.- Notes.- References.- 5 Pisot numbers, Salem numbers and distribution modulo 1.- 5.0 Notation.- 5.1 Some sequences (??n) non-uniformly distributed modulo 1.- 5.2 Pisot numbers and Salem numbers. Definitions and algebraic properties.- 5.3 Distribution modulo 1 of the sequences (?n) with ? a U-number.- 5.4 Pisot numbers and distribution modulo 1 of certain sequences (??n).- 5.5 Salem numbers and distribution modulo 1 of certain sequences (??n).- 5.6 Sequences (??n) non-uniformly distributed modulo 1.- Notes.- References.- 6 Limit points of Pisot and Salem sets.- 6.0 Notation.- 6.1 Closure of the set S.- 6.2 The derived set S? of S.- 6.3 Successive derived sets of S.- 6.4 Limit points of the set T.- Notes.- References.- 7 Small Pisot numbers.- 7.1 Schur's approximations forelements of N*1.- 7.2 Small Pisot numbers.- 7.3 The smallest number of S?.- Notes.- References.- 8 Some properties and applications of Pisot numbers.- 8.1 Some algebraic properties and applications of Pisot and Salem numbers.- 8.2 An application of Pisot numbers to a problem of uniform distribution.- 8.3 Application of Pisot numbers to a problem of rational approximations of algebraic numbers.- 8.4 Pisot numbers and the Jacobi-Perron algorithm.- Notes.- References.- 9 Algebraic number sets.- 9.1 Sq sets.- 9.2 n-tuples of algebraic numbers.- Notes.- References.- 10 Rational functions over rings of adeles.- 10.1 Adeles of Q.- 10.2 Analytic functions in Cp.- 10.3 Rationality criteria in QI[[X]].- 10.4 Compact families of rational functions.- Notes.- References.- 11 Generalizations of Pisot and Salem numbers to adeles.- 11.1 Definition of the set UI.- 11.2 Subsets of UI and characterizations.- 11.3 The sets SI?.- 11.4 The sets TI.- 11.5 The sets SIJ.- 11.6 The sets BI.- 11.7 Closed subsets of SI?.- 11.8 Limit points of the sets TI.- Notes.- References.- 12 Pisot elements in a field of formal power series.- 12.0 Generalities and notation.- 12.1 Definitions of the sets U and S.- 12.2 Characterizations of the sets U and S.- 12.3 Limit points of the sets U and S.- 12.4 Relation between the sets S and S.- Notes.- References.- 13 Pisot sequences, Boyd sequences and linear recurrence.- 13.0 Convergence theorems.- 13.1 Pisot sequences.- 13.2 Linear recurrence and Pisot sequences.- 13.3 Boyd sequences.- Notes.- References.- 14 Generalizations of Pisot and Boyd sequences.- 14.1 Convergence theorems in AI.- 14.2 Pisot sequences in AI.- 14.3 Boyd sequences in AI.- 14.4 Pisot and Boyd sequences in a field of formal power series.- Notes.- References.- l5 The Salem-Zygmundtheorem.- 15.1 Introduction.- 15.2 Sets of uniqueness.- 15.3 Symmetric perfect sets.- 15.4 The sufficient condition for the Salem-Zygmund theorem.- 15.5 A theorem by Senge and Strauss.- References.
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