JournalsrmiVol. 21, No. 1pp. 133–161

A multiple set version of the 3k33k-3 Theorem

  • Yahya ould Hamidoune

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

    Ecole Polytechnique, Palaiseau, France
A multiple set version of the $3k-3$ Theorem cover
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Abstract

In 1959, Freiman demonstrated his famous 3k43k-4 Theorem which was to be a cornerstone in inverse additive number theory. This result was soon followed by a 3k33k-3 Theorem, proved again by Freiman. This result describes the sets of integers A\mathcal{A} such that A+A3A3| \mathcal{A}+\mathcal{A} | \leq 3 | \mathcal{A} | -3. In the present paper, we prove a 3k33k-3-like Theorem in the context of multiple set addition and describe, for any positive integer jj, the sets of integers A\mathcal{A} such that the inequality jAj(j+1)(A1)/2|j \mathcal{A} | \leq j(j+1)(| \mathcal{A} | -1)/2 holds. Freiman's 3k33k-3 Theorem is the special case j=2j=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 jAj\mathcal{A}.

Cite this article

Yahya ould Hamidoune, Alain Plagne, A multiple set version of the 3k33k-3 Theorem. Rev. Mat. Iberoam. 21 (2005), no. 1, pp. 133–161

DOI 10.4171/RMI/418