Nathanson M.B. Additive Number Theory: Inverse Problems and the Geometry of Sumsets

Springer, 1996. — xiv, 296 p. — (Graduate Texts in Mathematics, 165). — ISBN 0-387-94655-1.Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer h > 2, and tries to describe the structure of the sumset hA consisting of all sums of h elements of A. By contrast, in an inverse problem, one starts with a sumset hA, and attempts to describe the structure of the underlying set A. In recent years there has been ramrkable progress in the study of inverse problems for finite sets of integers. In particular, there are important and beautiful inverse theorems due to Freiman, Kneser, Plünnecke, Vosper, and others. This volume includes their results, and culminates with an elegant proof by Ruzsa of the deep theorem of Freiman that a finite set of integers with a small sumset must be a large subset of an n-dimensional arithmetic progression.Notation
Simple inverse theorems
Direct and inverse problems
Finite arithmetic progressions
An inverse problem for distinct summands
A special case
Small sumsets: The case |2A| ≤ 3k-4
Application: The number of sums and products
Application: Sumsets and powers of 2
Notes
Exercises
Sums of congruence classes
Addition in groups
The e-transform
The Cauchy-Davenport theorem
The Erdös-Ginzburg-Ziv theorem
Vosper's theorem
Application: The range of a diagonal form
Exponential sums
The Freiman-Vosper theorem
Notes
Exercises
Sums of distinct congruence classes
The Erdos-Heilbronn conjecture
Vandermonde determinants
Multidimensional ballot numbers
A review of linear algebra
Alternating products
Erdös-Heilbronn, concluded
The polynomial method
Erdös-Heilbronn via polynomials
Notes
Exercises
Kneser's theorem for groups
Periodic subsets
The addition theorem
Application: The sum of two sets of integers
Application: Bases for finite and σ-finite groups
Notes
Exercises
Sums of vectors in Euclidean space
Small sumsets and hyperplanes
Linearly independent hyperplanes
Blocks
Proof of the theorem
Notes
Exercises
Geometry of numbers
Lattices and determinants
Convex bodies and Minkowski's First Theorem
Application: Sums of four squares
Successive minima and Minkowski's second theorem
Bases for sublattices
Torsion-free abelian groups
An important example
Notes
Exercises
Pliinnecke's inequality
Plunnecke graphs
Examples of Plunnecke graphs
Multiplicativity of magnification ratios
Menger's theorem
Plunnecke's inequality
Application: Estimates for sumsets in groups
Application: Essential components
Notes
Exercises
Freiman's theorem
Multidimensional arithmetic progressions
Freiman isomorphisms
Bogolyubov's method
Ruzsa's proof, concluded
Notes
Exercises
Applications of Freiman's theorem
Combinatorial number theory
Small sumsets and long progressions
The regularity lemma
The Balog-Szemerédi theorem
A conjecture of Erdös
The proper conjecture
Notes
Exercises
References
IndexФайл: отскан. стр. (b/w 600 dpi) + OCR + закладки