Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Prerequisites and notation
- 1 Introduction
- 2 The principle of the large sieve
- 3 Group and conjugacy sieves
- 4 Elementary and classical examples
- 5 Degrees of representations of finite groups
- 6 Probabilistic sieves
- 7 Sieving in discrete groups
- 8 Sieving for Frobenius over finite fields
- Appendix A Small sieves
- Appendix B Local density computations over finite fields
- Appendix C Representation theory
- Appendix D Property (T) and Property (τ)
- Appendix E Linear algebraic groups
- Appendix F Probability theory and random walks
- Appendix G Sums of multiplicative functions
- Appendix H Topology
- References
- Index
Preface
Published online by Cambridge University Press: 05 October 2009
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Prerequisites and notation
- 1 Introduction
- 2 The principle of the large sieve
- 3 Group and conjugacy sieves
- 4 Elementary and classical examples
- 5 Degrees of representations of finite groups
- 6 Probabilistic sieves
- 7 Sieving in discrete groups
- 8 Sieving for Frobenius over finite fields
- Appendix A Small sieves
- Appendix B Local density computations over finite fields
- Appendix C Representation theory
- Appendix D Property (T) and Property (τ)
- Appendix E Linear algebraic groups
- Appendix F Probability theory and random walks
- Appendix G Sums of multiplicative functions
- Appendix H Topology
- References
- Index
Summary
‘The Romans,’ Roger and the Reverend Dr. Paul de la Nuit were drunk together one night, or the vicar was, ‘the ancient Roman priests laid a sieve in the road, and then waited to see which stalks of grass would come up through the holes.’
Thomas Pynchon, ‘Gravity's Rainbow’These notes arose, by the long and convoluted process that research often turns out to be, from a supposedly short addition to my paper [80]. This is a story that is certainly typical of much of scientific research, and since I always find this fascinating, and hardly visible from the outside once a paper or book is published, I will summarize the events briefly. Readers who like science rather dry or dour may wish to start reading Chapter 1.
The original ambition was simply to extend the large sieve bound for Frobenius conjugacy classes of this first paper to the stronger form classically due to Montgomery, which would mean that ‘small sieve’ applications would become possible. The possibility of this extension seemed clear to me, as well as the relative paucity of new applications. At the same time, it seemed natural to ‘axiomatize’ the setting in a way allowing an identical treatment of the classical large sieve inequality and this newer variant, and this seemed a worthwhile enough goal.
All this should not have taken very long, either in time or space, except that inevitable delays due to teaching and other duties led to the thought that maybe other applications of this abstract form of sieve would be possible, and could be briefly discussed in the course of the paper, which would thus become stronger.
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- Information
- The Large Sieve and its ApplicationsArithmetic Geometry, Random Walks and Discrete Groups, pp. xi - xvPublisher: Cambridge University PressPrint publication year: 2008