# A multiple set version of the $3k−3$ Theorem

### Yahya ould Hamidoune

Université Pierre et Marie Curie, Paris, France### Alain Plagne

Ecole Polytechnique, Palaiseau, France

## Abstract

In 1959, Freiman demonstrated his famous $3k−4$ Theorem which was to be a cornerstone in inverse additive number theory. This result was soon followed by a $3k−3$ Theorem, proved again by Freiman. This result describes the sets of integers $A$ such that $∣A+A∣≤3∣A∣−3$. In the present paper, we prove a $3k−3$-like Theorem in the context of multiple set addition and describe, for any positive integer $j$, the sets of integers $A$ such that the inequality $∣jA∣≤j(j+1)(∣A∣−1)/2$ holds. Freiman's $3k−3$ Theorem is the special case $j=2$ 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 $jA$.

## Cite this article

Yahya ould Hamidoune, Alain Plagne, A multiple set version of the $3k−3$ Theorem. Rev. Mat. Iberoam. 21 (2005), no. 1, pp. 133–161

DOI 10.4171/RMI/418