In 1959, Freiman demonstrated his famous Theorem which was to be a cornerstone in inverse additive number theory. This result was soon followed by a Theorem, proved again by Freiman. This result describes the sets of integers such that . In the present paper, we prove a -like Theorem in the context of multiple set addition and describe, for any positive integer , the sets of integers such that the inequality holds. Freiman's Theorem is the special case of our result. This result implies, for example, the best known results on a function related to the Diophantine Frobenius number. Actually, our main theorem follows from a more general result on the border of .
Cite this article
Yahya ould Hamidoune, Alain Plagne, A multiple set version of the Theorem. Rev. Mat. Iberoam. 21 (2005), no. 1, pp. 133–161DOI 10.4171/RMI/418