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A multiple set version of the \(3k-3\) theorem. (English) Zbl 1078.11059
The authors introduce to Freiman’s inverse theorems and prove the following theorem, which is a classification of sets with small sumsets: Let \(A\) be a set of non-negative integers with \(\min A=0, \max A=M\),and \(\gcd A=1\). Let \(j\geq 2\) be an integer. Let \(jA\) denote the \(j\)-fold sumset. If the upper bound \(| jA | \leq \frac{j(j+1)}{2} ( | A| -1)\) holds, then one of the following happens:
\(| A | \geq 1+ \frac{M}{j}\).
\(A\) is the union of two arithmetic progressions with the same common difference.
\(A\) is an arithmetic progression modulo \(M\).
\(M\) is even and \(A\) is of the form \(A=\{0,M/2,M,x,x+M/2, 2x\}\) for some positive integer \(x<M/2\).
The special case \(j=2\) is a famous theorem by G. A. Freiman [Izv. Vyssh. Uchebn. Zaved., Mat. 1959, No. 6(13), 202–213 (1959; Zbl 0096.25904)]. Freiman’s theorem is also called the \(3k-3\) theorem; the third alternative does not occur in this case.
The paper concludes with an application to the Frobenius problem.

11P70 Inverse problems of additive number theory, including sumsets
11B75 Other combinatorial number theory
Full Text: DOI EuDML
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