Diophantine Approximations

Description

This self-contained treatment covers basic results on homogeneous approximation of real numbers; the analogue for complex numbers; basic results for nonhomogeneous approximation in the real case; the analogue for complex numbers; and fundamental properties of the multiples of an irrational number, for both fractional and integral parts. 1963 edition. TITLE......Page 3 PREFACE......Page 5 CONTENTS......Page 7 1.1 The pigeon-hole principle......Page 9 1.2. The theorem of Hurwitz......Page 11 1.3. Asymmetric approximation......Page 17 1.4. Further results......Page 21 2.1. The Minkowski results......Page 24 2.2. Further results......Page 29 3.1. A sequence of rational approximations to an irrational number......Page 31 3.2. The uniform distribution of the fractional parts......Page 32 3.3. The uniform distribution of the integral parts......Page 35 3.4. Kronecker's Theorem......Page 36 3.5. Results of Skolem and Bang......Page 42 3.6. Sets defined by rational numbers......Page 46 3.7. Further results......Page 52 4.1. Ford's theorem......Page 54 4.2. Further results......Page 61 5.1. The covering of lattice points......Page 62 5.2. A n inequality in the complex plane......Page 64 5.3. Minimum values of forms......Page 68 5.4. Further results......Page 69 Bibliography......Page 71 Index......Page 75