Many classical problems in additive number theory are direct problems, in which one starts with a setAof natural numbers and an integerH -& 2, and tries to describe the structure of the sumsethAconsisting of all sums ofhelements ofA. By contrast, in an inverse problem, one starts with a sumsethA, and attempts to describe the structure of the underlying setA. 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 ann-dimensional arithmetic progression.
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