The theory of sets of multiples, a subject that lies at the intersection of analytic and probabilistic number theory, has seen much development since the publication of "Sequences" by Halberstam and Roth nearly thirty years ago. The area is rich in problems, many of them still unsolved or arising from current work. In this book, the author gives a coherent, self-contained account of the existing theory, bringing the reader to the frontiers of research. One of the fascinations of the theory is the variety of methods applicable to it, which include Fourier analysis, group theory, high and ultra-low moments, probability and elementary inequalities, and several branches of number theory.
Author(s): Richard R. Hall
Series: Cambridge Tracts in Mathematics
Publisher: CUP
Year: 1996
Language: English
Pages: 280
Contents......Page 7
Preface......Page 9
Introduction......Page 11
Notation......Page 15
0.2 Sets of multiples and primitive sequences......Page 17
0.3 Densities......Page 19
0.4 The Heilbronn-Rohrbach and Behrend inequalities......Page 29
0.5 Total decomposition sets......Page 38
1.2 Erdos' criterion......Page 42
1.3 Behrend sequences......Page 52
1.4 Witnesses......Page 76
2.1 Introduction......Page 82
2.2 Upper bounds for tk(sad)......Page 84
2.3 Generalized Behrend inequalities......Page 98
2.4 Multilinear functions......Page 104
2.5 Formulae for the densities tk(d)......Page 107
3.1 Introduction......Page 112
3.2 A first lower bound for (sd, d)......Page 121
3.3 Upper bounds for (sad, sad)......Page 128
3.4 Primitive d......Page 133
3.5 Perfect sequences......Page 138
4.1 Introduction......Page 142
4.2 The Erdos-Renyi theorem: first variant......Page 146
4.3 The Erdos-Renyi theorem: second variant......Page 162
4.4 The Behrend sequences Q*(t)......Page 169
4.5 A conjecture of Erdos and Renyi......Page 183
5.1 Introduction......Page 186
5.2 Necessary and sufficient conditions......Page 195
5.3 Slowly switching sequences......Page 208
5.4 Slowly switching sequences: a reformulation......Page 214
6.1 Introduction......Page 219
6.2 Applications of divisor density......Page 221
6.3 Weyl sums and additive functions......Page 229
6.4 Weyl sums and discrepancy......Page 234
6.5 Lower bounds for discrepancy......Page 242
7.1 Introduction......Page 255
7.2 Short intervals......Page 256
7.3 The asymptotic formula for E Rk......Page 260
7.4 The asymptotic formula for H(x, y, z)......Page 266
Bibliography......Page 274
Index......Page 279
