This book deals with the solutions of a group of questions related both to the general theory of transcendental numbers and to the metrical theory of diophantine (and also algebraic) approximations. The fundamental problem in this field has been known in the literature since 1932 as Mahler's conjecture. The main result of this book is a proof of Mahler's conjecture and some analogous theorems.
In Part I, the "Classical" case of Mahler's conjecture, dealing with real and complex numbers, is considered. This part should be comprehensible to any who knows the elements of measure theory and possesses sufficient perseverance in over-coming purely logical difficulties. Part II is concerned with locally compact fields with nonarchimedean valuation. This part requires a general familiarity with the structure of fields with nonarchimedean valuation. All the necessary information is given in the text with references to the sources.
Author(s): V. G. Sprindžuk
Series: Translations of Mathematical Monographs, Vol. 25
Publisher: American Mathematical Society
Year: 1969
Language: English
Pages: C, viii+192, B
Tags: Математика;Теория чисел;
Cover
MAHLER'S PROBLEM IN METRIC NUMBER THEORY
Copyright
1969 by the American Mathematical Society
Library of Congress Card Number 73-86327
Standard Book Number 821-81575-X
PREFACE
TABLE OF CONTENTS
INTRODUCTION
§i. BASIC CONCEPTS
§2. HISTORICAL SURVEY
§3. GENERAL OUTLINE OF THE PROOF
Part I REAL AND COMPLEX NUMBERS
CHAPTER 1 AUXILIARY CONSIDERATIONS
§1. NOTATION
§2. LEMMAS ON POLYNOMIALS
§3. LEMMAS ON MEASURABLE SETS
§4. INVARIANCE OF THE PARAMETERS wn (ce )
§5. REDUCTION TO IRREDUCIBLE POLYNOMIALS
§6. REDUCTION TO THE POLYNOMIALS FROM Pn
§7. THE SIMPLEST SPECIAL CASES OF THE CONJECTURE
§8. THE EQUATION 02 = 1
CHAPTER 2 THE COMPLEX CASE
§1. THE DOMAINS a(P)
§2. INESSENTIAL DOMAINS
§3. DECOMPOSITION INTO E-CLASSES
§4. REDUCTION' TO THE ROOTS OF A, FIXED CLASS K (r)
§5. CLASSES OF THE FIRST KIND
§6. CLASSES OF THE SECOND KIND
§7. CONCLUSION OF THE PROOF
CHAPTER 3 THE REAL CASE
§1. DECOMPOSITION INTO c-CL ASSE S
§2. CLASSES OF THE FIRST KIND
§3. CLASSES OF THE SECOND KIND
§4. CONCLUSION OF THE PROOF
Part II FIELDS WITH NON-ARCHIMEDEAN VALUATION
CHAPTER1 BASIC FACTS
§1. INTRODUCTION
§2. MEASURE ON A LOCALLY COMPACT FIELD
§3. PROPERTIES OF THE MEASURE
§4. DENSITY AND MEASURE
§5. A LEMMA ON PARTIAL COVERINGS
§6. A REMARK ON EXTENDING A VALUATION
§7. ESTIMATES FOR THE DISTANCE 1a) - K I
§8. THE STRUCTURE OF THE DOMAINS ai (F)
§9. CONCLUSION
CHAPTER 2 FIELDS OF p-ADIC NUMBERS
§1. DIOPHANTINE APPROXIMATION IN Qp
§2. LEMMAS ON POLYNOMIALS
§3. PRELIMINARY REMARKS
§4. REDUCTION TO THE POLYNOMIALS FROM Fn
§ 5. THE SIMPLEST SPECIAL CASES
§6. DECOMPOSITION INTO E-CLASSES
§7. REDUCTION TO THE ROOTS OF A FIXED CLASS
§8. INESSENTIAL DOMAINS
§9. ESSENTIAL DOMAINS
§10. CLASSES OF THE SECOND KIND
§11. CONCLUSION OF THE PROOF
CHAPTER 3 FIELDS OF FORMAL POWER SERIES
§1. NOTATION
§2. BASIC FACTS FROM THE "GEOMETRY OF NUMBERS"
§ 3. LEMMAS ON POLYNOMIALS
§4. PRELIMINARY REMARKS
§5. REDUCTION TO THE POLYNOMIALS FROM Pn
§6. THE SIMPLEST SPECIAL CASES
§7. DECOMPOSITION INTO E-CLASSES
§8. THE DOMAINS Qi (P)
§9. INESSENTIAL DOMAINS
§10. ESSENTIAL DOMAINS
§11. CLASSES OF THE SECOND KIND
§12. CONCLUSION OF THE PROOF
SUPPLEMENTARY RESULTS AND REMARKS
A. REAL AND COMPLEX NUMBERS
B. FIELDS OF p-ADIC NUMBERS
C. FIELDS OF POWER SERIES
CONCLUSION
§1. SOME COROLLARIES TO THE RESULTS OBTAINED
§2. GENERAL CONSEQUENCES
§3. NEW PROBLEMS AND CONJECTURES
AN APPLICATION
§1. SIMULTANEOUS LINEAR DIOPHANTINE APPROXIMATIONS
§2. SIMULTANEOUS APPROXIMATIQNS OF NUMBERSWITH A QUADRATIC RELATION
BIBLIOGRAPHY
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