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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.

##### MSC:
 11P70 Inverse problems of additive number theory, including sumsets 11B75 Other combinatorial number theory
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