《加性數論:逆問題與和集幾何》內容簡介：The rerequisites for this book are undergraduate courses in elementary number theory,algebra,and analysis.Beyond this,the volume is self-contained.I include complete proofs of results from exterior algebra,combinatorics,graph theory,and the geometry of numbers that are used in the proofs of the Erdos-Heilbronn
基本介紹
- 外文名：Additive Number Theory:Inverse Problens And The Geometry Of Sumsets
- 書名：加性數論:逆問題與和集幾何
- 作者：納森
- 出版日期：2012年6月1日
- 語種：簡體中文, 英語
- ISBN：9787510044083
- 品牌：世界圖書出版公司北京公司
- 出版社：世界圖書出版公司北京公司
- 頁數：291頁
- 開本：24
- 定價：45.00
內容簡介,圖書目錄,
內容簡介
《加性數論:逆問題與和集幾何》由世界圖書出版公司北京公司出版。
圖書目錄
Preface
Notation
1 Simple inverse theorems
1.1 Direct and inverse problems
1.2 Finite arithmetic progressions
1.3 An inverse problem for distinct summands
1.4 A special case
1.5 Small sumsets: The case |2A| ≤ 3k-4
1.6 Application: The number of sums and products
1.7 Application: Sumsets and powers of 2
1.8 Notes
1.9 Exercises
2 Sums of congruence classes
2.1 Addition in groups
2.2 The e-transform
2.3 The Cauchy-Davenport theorem
2.4 The Erdos-Ginzburg-Ziv theorem
2.5 Vosper's theorem
2.6 Application: The range of a diagonal form
2.7 Exponential sums
2.8 The Freiman-Vosper theorem
2.9 Notes
2.10 Exercises
3 Sums of distinct congruence classes
3.1 The Erdos-Heilbronn conjecture
3.2 Vandermonde determinants
3.3 Multidimensional ballot numbers
3.4 A review of linear algebra
3.5 Alternating products
3.6 Erdos-Heilbronn, concluded
3.7 The polynomial method
3.8 Erdos-Heilbronn via polynomials
3.9 Notes
3.10 Exercises
4 Kneser's theorem for groups
4.1 Periodic subsets
4.2 The addition theorem
4.3 Application: The sum of two sets of integers
4.4 Application: Bases for finite and o-finite groups
4.5 Notes
4.6 Exercises
5 Sums of vectors in Euclidean space
5.1 Small sumsets and hyperplanes
5.2 Linearly independent hyperplanes
5.3 Blocks
5.4 Proof of the theorem
5.5 Notes
5.6 Exercises
6 Geometry of numbers
6.1 Lattices and determinants
6.2 Convex bodies and Minkowski's First Theorem
6.3 Application: Sums of four squares
6.4 Successive minima and Minkowski's second theorem
6.5 Bases for sublattices
6.6 Torsion-free abelian groups
6.7 An important example
6.8 Notes
6.9 Exercises
7. Pliinnecke's inequality
7.1 Plunnecke graphs
7.2 Examples of Plunnecke graphs
7.3 Multiplicativity of magnification ratios
7.4 Menger's theorem
7.5 Pliinnecke's inequality
7.6 Application: Estimates for sumsets in groups
7.7 Application: Essential components
7.8 Notes
7.9 Exercises
8 Freiman's theorem
8.1 Multidimensional arithmetic progressions
8.2 Freiman isomorphisms
8.3 Bogolyubov's method
8.4 Ruzsa's proof, concluded
8.5 Notes
8.6 Exercises
9 Applications of Freiman's theorem
9.1 Combinatorial number theory
9.2 Small sumsets and long progressions
9.3 The regularity lemma
9.4 The Balog-Szemeredi theorem
9.5 A conjecture of Erdos
9.6 The proper conjecture
9.7 Notes
9.8 Exercises
References
Index
Notation
1 Simple inverse theorems
1.1 Direct and inverse problems
1.2 Finite arithmetic progressions
1.3 An inverse problem for distinct summands
1.4 A special case
1.5 Small sumsets: The case |2A| ≤ 3k-4
1.6 Application: The number of sums and products
1.7 Application: Sumsets and powers of 2
1.8 Notes
1.9 Exercises
2 Sums of congruence classes
2.1 Addition in groups
2.2 The e-transform
2.3 The Cauchy-Davenport theorem
2.4 The Erdos-Ginzburg-Ziv theorem
2.5 Vosper's theorem
2.6 Application: The range of a diagonal form
2.7 Exponential sums
2.8 The Freiman-Vosper theorem
2.9 Notes
2.10 Exercises
3 Sums of distinct congruence classes
3.1 The Erdos-Heilbronn conjecture
3.2 Vandermonde determinants
3.3 Multidimensional ballot numbers
3.4 A review of linear algebra
3.5 Alternating products
3.6 Erdos-Heilbronn, concluded
3.7 The polynomial method
3.8 Erdos-Heilbronn via polynomials
3.9 Notes
3.10 Exercises
4 Kneser's theorem for groups
4.1 Periodic subsets
4.2 The addition theorem
4.3 Application: The sum of two sets of integers
4.4 Application: Bases for finite and o-finite groups
4.5 Notes
4.6 Exercises
5 Sums of vectors in Euclidean space
5.1 Small sumsets and hyperplanes
5.2 Linearly independent hyperplanes
5.3 Blocks
5.4 Proof of the theorem
5.5 Notes
5.6 Exercises
6 Geometry of numbers
6.1 Lattices and determinants
6.2 Convex bodies and Minkowski's First Theorem
6.3 Application: Sums of four squares
6.4 Successive minima and Minkowski's second theorem
6.5 Bases for sublattices
6.6 Torsion-free abelian groups
6.7 An important example
6.8 Notes
6.9 Exercises
7. Pliinnecke's inequality
7.1 Plunnecke graphs
7.2 Examples of Plunnecke graphs
7.3 Multiplicativity of magnification ratios
7.4 Menger's theorem
7.5 Pliinnecke's inequality
7.6 Application: Estimates for sumsets in groups
7.7 Application: Essential components
7.8 Notes
7.9 Exercises
8 Freiman's theorem
8.1 Multidimensional arithmetic progressions
8.2 Freiman isomorphisms
8.3 Bogolyubov's method
8.4 Ruzsa's proof, concluded
8.5 Notes
8.6 Exercises
9 Applications of Freiman's theorem
9.1 Combinatorial number theory
9.2 Small sumsets and long progressions
9.3 The regularity lemma
9.4 The Balog-Szemeredi theorem
9.5 A conjecture of Erdos
9.6 The proper conjecture
9.7 Notes
9.8 Exercises
References
Index
