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An introduction to both Nevanlinna theory and Diophantine approximation, with emphasis on the analogy between these two subjects. Each chapter covers both subjects, and a table is provided at the end of each chapter to indicate the correspondence of theorems.

Diophantine approximation. --- Nevanlinna theory. --- Functions, Meromorphic --- Value distribution theory --- Approximation, Diophantine --- Approximation theory --- Diophantine analysis

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The book timely surveys new research results and related developments in Diophantine approximation, a division of number theory which deals with the approximation of real numbers by rational numbers. The book is appended with a list of challenging open problems and a comprehensive list of references. From the contents: Field extensions Algebraic numbers Algebraic geometry Height functions The abc-conjecture Roth's theorem Subspace theorems Vojta's conjectures L-functions.

Diophantine approximation. --- Nevanlinna theory. --- Functions, Meromorphic --- Value distribution theory --- Approximation, Diophantine --- Approximation theory --- Diophantine analysis --- Algebraic Geometry. --- Algebraic Numbers. --- Elliptic Curves.

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This volume contains 22 research and survey papers on recent developments in the field of diophantine approximation. The first article by Hans Peter Schlickewei is devoted to the scientific work of Wolfgang Schmidt. Further contributions deal with the subspace theorem and its applications to diophantine equations and to the study of linear recurring sequences. The articles are either in the spirit of more classical diophantine analysis or of geometric or combinatorial flavor. In particular, estimates for the number of solutions of diophantine equations as well as results concerning congruences and polynomials are established. Furthermore, the volume contains transcendence results for special functions and contributions to metric diophantine approximation and to discrepancy theory. The articles are based on lectures given at a conference at the Erwin Schr6dinger Institute in Vienna in 2003, in which many leading experts in the field of diophantine approximation participated. The editors are very grateful to the Erwin Schr6dinger Institute and to the FWF (Austrian Science Fund) for the financial support and they express their particular thanks to Springer-Verlag for the excellent cooperation. Robert E Tichy Diophantine Approximation H. E Schlickewei et al. , Editors 9 Springer-Verlag 2008 THE MATHEMATICAL WORK OF WOLFGANG SCHMIDT Hans Peter Schlickewei Mathematik Informatik, und Philipps-Universitiit Hans-Meerwein-Strasse, Marburg, 35032 Marburg, Germany k. Introduction Wolfgang Schmidt's mathematical activities started more than fifty years ago in 1955. In the meantime he has written more than 180 papers - many of them containing spectacular results and breakthroughs in different areas of number theory.

Diophantine approximation --- Approximation, Diophantine --- Approximation theory --- Diophantine analysis --- Algebra. --- Computer science --- Number theory. --- Computational Mathematics and Numerical Analysis. --- Number Theory. --- Mathematics. --- Number study --- Numbers, Theory of --- Algebra --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematics --- Mathematical analysis --- Computer mathematics.

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This book gives a comprehensive treatment of random phenomena and distribution results in diophantine approximation, with a particular emphasis on quadratic irrationals. It covers classical material on the subject as well as many new results developed by the author over the past decade. A range of ideas from other areas of mathematics are brought to bear with surprising connections to topics such as formulae for class numbers, special values of L-functions, and Dedekind sums. Care is taken to elaborate difficult proofs by motivating major steps and accompanying them with background explanations, enabling the reader to learn the theory and relevant techniques. Written by one of the acknowledged experts in the field, Probabilistic Diophantine Approximation is presented in a clear and informal style with sufficient detail to appeal to both advanced students and researchers in number theory.

Diophantine approximation. --- Approximation, Diophantine --- Approximation theory --- Diophantine analysis --- Number theory. --- Distribution (Probability theory. --- Number Theory. --- Probability Theory and Stochastic Processes. --- Number study --- Numbers, Theory of --- Algebra --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk

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The second edition of the book includes a new chapter on the study of composition operators on the Hardy space and their complete characterization by Gordon and Hedenmalm. The book is devoted to Diophantine approximation, the analytic theory of Dirichlet series and their composition operators, and connections between these two domains which often occur through the Kronecker approximation theorem and the Bohr lift. The book initially discusses Harmonic analysis, including a sharp form of the uncertainty principle, Ergodic theory and Diophantine approximation, basics on continued fractions expansions, and the mixing property of the Gauss map and goes on to present the general theory of Dirichlet series with classes of examples connected to continued fractions, Bohr lift, sharp forms of the Bohnenblust–Hille theorem, Hardy–Dirichlet spaces, composition operators of the Hardy–Dirichlet space, and much more. Proofs throughout the book mix Hilbertian geometry, complex and harmonic analysis, number theory, and ergodic theory, featuring the richness of analytic theory of Dirichlet series. This self-contained book benefits beginners as well as researchers. .

Mathematics. --- Dynamics. --- Ergodic theory. --- Vibration. --- Dynamical systems. --- Mathematics, general. --- Dynamical Systems and Ergodic Theory. --- Vibration, Dynamical Systems, Control. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Cycles --- Sound --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Math --- Science --- Diophantine approximation. --- Approximation, Diophantine --- Approximation theory --- Diophantine analysis