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Intro; Preface; Chapter 2 by Michael Coons and Lukas Spiegelhofer Number Theoretic Aspects of Regular Sequences; Chapter 3 by Émilie CharlierFirst-Order Logic and Numeration Systems; Chapter 4 by Jason BellSome Applications of Algebra to Automatic Sequences; Chapter 5 by Pascal Ochem, Michaël Rao, and Matthieu RosenfeldAvoiding or Limiting Regularities in Words; Chapter 6 by Caïus Wojcik and Luca ZamboniColoring Problems for Infinite Words; Chapter 7 by Verónica Becher and Olivier CartonNormal Numbers and Computer Science; Chapter 8 by Manfred MadritschNormal Numbers and Symbolic Dynamics
Chapter 9 by Nathalie Aubrun, Sebastián Barbieri, and Emmanuel JeandelAbout the Domino Problem for Subshifts on GroupsChapter 10 by Ines Klimann and Matthieu Picantin Automaton (Semi)groups: Wang Tilings and Schreier Tries; Chapter 11 by Laurent BartholdiAmenability Groups and G-Sets; Acknowledgments; Contents; Contributors; 1 General Framework; 1.1 Conventions; 1.2 Algebraic Structures; 1.3 Words; 1.3.1 Finite Words; 1.3.2 Infinite Words; 1.3.3 Number Representations; 1.3.4 Normality; 1.3.5 Repetitions in Words; 1.4 Morphisms; 1.5 Languages and Machines; 1.5.1 Languages of Finite Words
1.5.2 Formal Series1.5.3 Codes; 1.5.4 Automata; 1.6 Sequences and Machines; 1.6.1 Automatic Sequences; 1.6.2 Regular Sequences; 1.7 Dynamical Systems; 1.7.1 Topological Dynamical Systems; 1.7.2 Measure-Theoretic Dynamical Systems; 1.7.3 Symbolic Dynamics; 2 Number Theoretic Aspects of Regular Sequences; 2.1 Introduction; 2.1.1 Two Important Questions; 2.1.2 Three (or Four) Hierarchies in One; 2.2 From Automatic to Regular to Mahler; 2.2.1 Definitions; 2.2.2 Some Comparisons Between Regular and Mahler Functions; 2.3 Size and Growth; 2.3.1 Lower Bounds; 2.3.2 Upper Bounds
2.3.3 Maximum Values and the Finiteness Property2.4 Analytic and Algebraic Properties of Mahler Functions; 2.4.1 Analytic Properties of Mahler Functions; 2.4.2 Rational-Transcendental Dichotomy of Mahler Functions; 2.5 Rational-Transcendental Dichotomy of Regular Numbers; 2.6 Diophantine Properties of Mahler Functions; 2.6.1 Rational Approximation of Mahler Functions; 2.6.2 A Transcendence Test for Mahler Functions; 2.6.3 Algebraic Approximation of Mahler Functions; 3 First-Order Logic and Numeration Systems; 3.1 Introduction; 3.2 Recognizable Sets of Nonnegative Integers
3.2.1 Unary Representations3.2.2 Integer Bases; 3.2.3 Positional Numeration Systems; 3.2.4 Abstract Numeration Systems; 3.2.5 The Cobham-Semenov Theorem; 3.3 First-Order Logic and b-Automatic Sequences; 3.3.1 b-Definable Sets of Integers; 3.3.2 The Büchi-Bruyère Theorem; 3.3.3 The First-Order Theory of ""426830A N,+,Vb""526930B Is Decidable; 3.3.4 Applications to Decidability Questions for b-Automatic Sequences; 3.4 Enumeration; 3.4.1 b-Regular Sequences; 3.4.2 N-Recognizable and N∞-RecognizableFormal Series; 3.4.3 Counting b-Definable Properties of b-Automatic Sequences Is b-Regular
Chapter 9 by Nathalie Aubrun, Sebastián Barbieri, and Emmanuel JeandelAbout the Domino Problem for Subshifts on GroupsChapter 10 by Ines Klimann and Matthieu Picantin Automaton (Semi)groups: Wang Tilings and Schreier Tries; Chapter 11 by Laurent BartholdiAmenability Groups and G-Sets; Acknowledgments; Contents; Contributors; 1 General Framework; 1.1 Conventions; 1.2 Algebraic Structures; 1.3 Words; 1.3.1 Finite Words; 1.3.2 Infinite Words; 1.3.3 Number Representations; 1.3.4 Normality; 1.3.5 Repetitions in Words; 1.4 Morphisms; 1.5 Languages and Machines; 1.5.1 Languages of Finite Words
1.5.2 Formal Series1.5.3 Codes; 1.5.4 Automata; 1.6 Sequences and Machines; 1.6.1 Automatic Sequences; 1.6.2 Regular Sequences; 1.7 Dynamical Systems; 1.7.1 Topological Dynamical Systems; 1.7.2 Measure-Theoretic Dynamical Systems; 1.7.3 Symbolic Dynamics; 2 Number Theoretic Aspects of Regular Sequences; 2.1 Introduction; 2.1.1 Two Important Questions; 2.1.2 Three (or Four) Hierarchies in One; 2.2 From Automatic to Regular to Mahler; 2.2.1 Definitions; 2.2.2 Some Comparisons Between Regular and Mahler Functions; 2.3 Size and Growth; 2.3.1 Lower Bounds; 2.3.2 Upper Bounds
2.3.3 Maximum Values and the Finiteness Property2.4 Analytic and Algebraic Properties of Mahler Functions; 2.4.1 Analytic Properties of Mahler Functions; 2.4.2 Rational-Transcendental Dichotomy of Mahler Functions; 2.5 Rational-Transcendental Dichotomy of Regular Numbers; 2.6 Diophantine Properties of Mahler Functions; 2.6.1 Rational Approximation of Mahler Functions; 2.6.2 A Transcendence Test for Mahler Functions; 2.6.3 Algebraic Approximation of Mahler Functions; 3 First-Order Logic and Numeration Systems; 3.1 Introduction; 3.2 Recognizable Sets of Nonnegative Integers
3.2.1 Unary Representations3.2.2 Integer Bases; 3.2.3 Positional Numeration Systems; 3.2.4 Abstract Numeration Systems; 3.2.5 The Cobham-Semenov Theorem; 3.3 First-Order Logic and b-Automatic Sequences; 3.3.1 b-Definable Sets of Integers; 3.3.2 The Büchi-Bruyère Theorem; 3.3.3 The First-Order Theory of ""426830A N,+,Vb""526930B Is Decidable; 3.3.4 Applications to Decidability Questions for b-Automatic Sequences; 3.4 Enumeration; 3.4.1 b-Regular Sequences; 3.4.2 N-Recognizable and N∞-RecognizableFormal Series; 3.4.3 Counting b-Definable Properties of b-Automatic Sequences Is b-Regular