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Structural Additive Theory

Part of the Developments in Mathematics Book Series (volume 30)

DOI: 10.1007/978-3-319-00416-7

ISBN: 978-3-319-00415-0

Publisher: Springer International Publishing - Springer Journals

Apr 2, 2013

1 - 10 of 22 Chapters

LitStream

Chapter 1Abelian Groups and Character Sums

Grynkiewicz, David J.

Pages: 1 –10

[This first chapter should be viewed more as a reference chapter. The goal is to present several results concerning abelian groups that may have been missed in a standard first course in Group Theory. We begin first with basic notation used throughout the book, then some basic review leading up...
LitStream

Introduction to Sumsets

Grynkiewicz, David J.

Pages: 13 –23

[This chapter introduces the first main branch of study in the course: sumsets. We begin with all relevant definitions and notation, including those needed for our treatment of infinite summands. Arithmetic progressions and various important ways of decomposing sets are discussed in detail. A...
LitStream

Simple Results for Torsion-Free Abelian Groups

Grynkiewicz, David J.

Pages: 25 –28

[In this chapter, we prove two very basic results concerning finite sumsets over a torsion-free abelian group. First, we show that any finite sumset over a torsion-free abelian group is isomorphic to a sumset over the integers. Second, we give the basic lower bound for finite sumsets over a...
LitStream

Basic Results for Sumsets with an Infinite Summand

Grynkiewicz, David J.

Pages: 29 –56

[Recall that, when A and B are nonempty subsets of an abelian group G, we defined \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
LitStream

The Pigeonhole and Multiplicity Bounds

Grynkiewicz, David J.

Pages: 57 –60

[The goal of this chapter is to prove the following basic result.]
LitStream

Periodic Sets and Kneser’s Theorem

Grynkiewicz, David J.

Pages: 61 –69

[The goal of this chapter is to introduce and expound one of the most fundamental and useful theorems in Inverse Additive Theory: Kneser’s Theorem. As remarked before, we are interested in determining the structure of A and B with small sumset. We saw in the previous sections that taking A and B...
LitStream

Compression, Complements and the 3k−4 Theorem

Grynkiewicz, David J.

Pages: 71 –98

[We have seen (from Kneser’s Theorem) that if |A+B|<|A|+|B|−1, then A+B must be periodic, which may be viewed as a partial structural description of A+B. In this chapter, we will prove a more precise structure theorem describing \documentclass[12pt]{minimal} \usepackage{amsmath}...
LitStream

Additive Energy

Grynkiewicz, David J.

Pages: 99 –109

[We introduce the concept of additive energy, which has proved itself indispensable for studying additive problems. We then give two simple applications of the concept. One involves showing that a Sidon set must have large sumset with any other subset. The other is a classical result of Vosper....
LitStream

Kemperman’s Critical Pair Theory

Grynkiewicz, David J.

Pages: 111 –132

[The goal of this chapter is to prove one of the most important inverse results for an arbitrary abelian group—the Kemperman Structure Theorem (KST)—which determines all finite, nonempty subsets A, B⊆G with |A+B|≤|A|+|B|−1. As the proof is already involved enough for the case when A and B are...
LitStream

Zero-Sums, Setpartitions and Subsequence Sums

Grynkiewicz, David J.

Pages: 135 –144

[With this chapter, we turn to the second main topic of the course: Subsequence sums and zero-sums. We begin by introducing the modern algebraic language for describing sequences and their subsums, which will be used throughout the remainder of the book. The notion of a setpartition is also...

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