Metric Number Theory

by Harman, Glyn

• ISBN13: 9780198500834
• ISBN10: 0198500831
• Format: Hardcover
• Copyright: 1998-08-13
• Publisher: Clarendon Press

Summary

This book examines the number-theoretic properties of the real numbers. It collects a variety of new ideas and develops connections between different branches of mathematics. An indispensable compendium of basic results, the text also includes important theorems and open problems. The book begins with the classical results of Borel, Khintchine, and Weyl, and then proceeds to Diophantine approximation, GCD sums, Schmidt's method, and uniform distribution. Other topics include generalizations to higher dimensions and various non-periodic problems (for example, restricting approximation to fractions with prime numerator and denominator). It concludes with a chapter on Hausdorf dimensions for exceptional sets of measure zero.

Table of Contents

Notation	xiii	(2)
Introduction	xv
1 Normal numbers	1	(23)
1.1 Definitions and elementary properties	1	(6)
1.2 Metrical lemmas and Borel's theorem	7	(11)
1.3 The law of the iterated logarithm	18	(5)
Notes	23	(1)
2 Diophantine approximation	24	(36)
2.1 Statement of results	24	(5)
2.2 Zero-one laws	29	(8)
2.3 The Duffin and Schaeffer theorem	37	(7)
2.4 Vaaler's theorem	44	(7)
2.5 Proof of Theorems 2.3 and 2.8	51	(2)
2.6 The Duffin and Schaeffer conjecture reformulated	53	(5)
Notes	58	(2)
3 GCD sums with applications	60	(34)
3.1 Statement of results	60	(7)
3.2 Proof of Theorem 3.1	67	(3)
3.3 Proof of Theorem 3.2	70	(8)
3.4 Proof of Theorems 3.3 and 3.4	78	(4)
3.5 Proof of Theorem 3.5	82	(4)
3.6 Proof of Theorems 3.6 and 3.7	86	(5)
3.7 Proof of Theorem 3.8	91	(2)
Notes	93	(1)
4 Schmidt's method	94	(26)
4.1 Statement of results	94	(3)
4.2 Proof of Theorems 4.1 and 4.2	97	(8)
4.3 Proof of Theorem 4.3	105	(4)
4.4 Proof of Theorem 4.4	109	(3)
4.5 The metric theory of continued fractions	112	(3)
4.6 A generalization to higher dimensions	115	(3)
Notes	118	(2)
5 Uniform distribution	120	(44)
5.1 Definitions and elementary properties	120	(6)
5.2 Trigonometric sums, the Erdos-Turan theorem and the Weyl criterion	126	(5)
5.3 The metrical theory of uniform distribution	131	(20)
5.4 Uniform distribution in higher dimensions	151	(10)
Notes	161	(3)
6 Diophantine approximation with restricted numerator and denominator	164	(23)
6.1 Introduction and statement of results	164	(7)
6.2 Proof of Theorem 6.2	171	(4)
6.3 Proof of Theorem 6.3	175	(2)
6.4 Proof of Theorem 6.4	177	(2)
6.5 Proof of Theorems 6.5 and 6.6	179	(1)
6.6 Proof of Theorem 6.7	180	(6)
Notes	186	(1)
7 Non-integer sequences	187	(28)
7.1 Introduction and statement of results	187	(5)
7.2 Proof of Theorems 7.1 and 7.2	192	(6)
7.3 A reduction of the problem and proofs for Theorems 7.3 and 7.5	198	(4)
7.4 Proof of Theorem 7.4	202	(4)
7.5 Proof of Theorem 7.6	206	(5)
7.6 Proof of Theorems 7.7 and 7.8	211	(2)
Notes	213	(2)
8 The integer parts of sequences	215	(26)
8.1 Introduction and statement of results	215	(5)
8.2 Proof of Theorem 8.1	220	(6)
8.3 Proof of Theorem 8.2	226	(3)
8.4 Proof of Theorem 8.3	229	(4)
8.5 Proof of Theorem 8.4	233	(1)
8.6 Proof of Theorem 8.5	234	(3)
8.7 Proof of Theorem 8.6	237	(1)
8.8 Proof of Theorem 8.7	238	(2)
Notes	240	(1)
9 Diophantine approximation on manifolds	241	(21)
9.1 Introduction	241	(4)
9.2 Proof of Theorem 9.2	245	(11)
9.3 Proof of Theorem 9.3	256	(5)
Notes	261	(1)
10 Hausdorff dimension of exceptional sets	262	(18)
10.1 Introduction and statement of results	262	(4)
10.2 Proof of Theorems 10.1 and 10.2	266	(1)
10.3 Proof of Theorems 10.3 and 10.4	267	(4)
10.4 Proof of Theorems 10.5, 10.6, and 10.7	271	(5)
10.5 Proof of Theorem 10.8	276	(2)
Notes	278	(2)
References	280	(15)
Index	295