加性數(shù)論

內(nèi)容概要

　　《加性數(shù)論：逆問題與和集幾何》分為上下2卷。堆壘數(shù)論討論的是很經(jīng)典的直接問題。在這個問題中，首先假定有一個自然數(shù)集合a和大于等于2的整數(shù)h，定義的和集ha是由所有的h和a中元素乘積的和組成，試圖描述和集ha的結(jié)構(gòu)；相反地，在逆問題中，從和集ha開始，去尋找這樣的一個集合a。近年來，有關(guān)整數(shù)有限集的逆問題方面取得了顯著進展。特別地，freiman，
kneser， plünnecke，
vosper以及一些其他的學(xué)者在這方面做出了突出的貢獻?！都有詳?shù)論：逆問題與和集幾何》中包括了這些結(jié)果，并且用freiman定理的ruzsa證明將《加性數(shù)論：逆問題與和集幾何》的內(nèi)容推向了高潮。
　　《加性數(shù)論：逆問題與和集幾何》讀者對象：數(shù)學(xué)專業(yè)的研究生和相關(guān)專業(yè)的科研人員。

書籍目錄

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 erd6s-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 erd6s-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 a-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. plunnecke'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 erd6s
9.6 the proper conjecture
9.7 notes
9.8 exercises
references
index

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