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March, 2005 A multiple set version of the $3k-3$ Theorem
Yahya ould Hamidoune, Alain Plagne
Rev. Mat. Iberoamericana 21(1): 133-161 (March, 2005).


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 $\mathcal{A}$ such that $| \mathcal{A}+\mathcal{A} | \leq 3 | \mathcal{A} | -3$. 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 $\mathcal{A}$ such that the inequality $|j \mathcal{A} | \leq j(j+1)(| \mathcal{A} | -1)/2$ 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 $j\mathcal{A}$.


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Yahya ould Hamidoune. Alain Plagne. "A multiple set version of the $3k-3$ Theorem." Rev. Mat. Iberoamericana 21 (1) 133 - 161, March, 2005.


Published: March, 2005
First available in Project Euclid: 22 April 2005

zbMATH: 1078.11059
MathSciNet: MR2155017

Primary: 11B75 , 11P70

Keywords: $3k-3$ theorem , $3k-4$ theorem , Frobenius problem , multiple set addition , structure theory of set addition

Rights: Copyright © 2005 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.21 • No. 1 • March, 2005
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