RÉSUMÉ

Let A be a set of k integers. G. Freiman's work done nearly half century ago on the set addition started the revelation of a general phenomenon for inverse problems. He showed that if the cardinality of A+A is small, then A must have some arithmetic structure. Freiman's theorems can be divided into two groups. The theorems in the first group characterize the structure of A when the cardinality of A+A is less than or equal to 3k-2. The theorems in the second group characterize the structure of A when the cardinality of A+A is less than ck, where c is an arbitrary constant independent of k. Although the theorems in the second group are more general, the theorems in the first group give more accurate structural characterization. Since then many generalizations and new developments have been obtained. For example, Y. O. Hamidoune and A. Plagne in a 2002 paper characterize the structure of A when the cardinality of A - A is 3k-3. However, there has been no advance for generalizing the theorems in the first group when the cardinality of A+A is greater than 3k-2 until now. In the talk the speaker will present his recent result of characterizing the structure of A when k is large enough and the cardinality of A+A is 3k-3+b where the positive number b is bounded by a given positive function f satisfying f(k) = o(k). This confirms a weak version of a Freiman's conjecture made five years ago. In the talk the speaker will also explain how the methods from nonstandard analysis play a crucial role in the proof.

WWW page design par Thomas Papanikolaou & Christine Bachoc