DIMACS TR: 93-33

Inverse Theorems in Additive Number Theory



Author: Melvyn B. Nathanson

ABSTRACT

The subject of this book is inverse theorems in additive number theory. The major results are due to Vosper, Kneser, and Freiman. The goal of the chapters in this preprint is to give a self-contained account of Ruzsa's proof of Freiman's inverse theorem. The only prerequisite is elementary number theory. I include proofs of such ``standard'' results as Menger's theorem in graph theory and Minkowski's theorem on successive minima in the geometry of numbers.

This is a preliminary version of the text. Some chapters are not yet written, or are incomplete. Other chapters have not been checked, and contain typographical and mathematical errors. I ask the reader to be patient with the manuscript, and to send me comments and corrections, both substantive and typographical, no matter how trivial. My e-mail address is: nathansn@dimacs.rutgers.edu.

I wish to thank Imre Z. Ruzsa for providing me with preprints of his papers on Freiman's theorem, Endre Szemer\'edi for describing to me his joint result with Antal Balog, and Gregory Freiman for valuable discussions about his work.

Paper available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1993/93-33.ps

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