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Some results in additive number theory. I: The critical pair theory. (English) Zbl 0985.11011
Let $$U\subset \mathbb N$$ be a finite set with $$\text{gcd}(U)=1$$ and $$m= \max(U)$$, $$n=\min(U).$$ The Frobenius number $$G(U)$$ is the maximal integer that cannot be expressed as a sum of elements of $$U$$. Let $$G$$ be an abelian group, $$A$$, $$B$$ be finite subsets of $$G$$, with $$|A|,|B|\geq 2.$$ Many authors investigated the cardinality of $$A+B$$ in comparison with $$|G|$$.
In the present paper the author investigates conditions for the validity of the inequality $|A+B|\geq \min\{|G|-1,|A|+|B|-1\}.$ As an application, using a new method, the author proves bounds on the Frobenius number. A conjecture of Lewin is proved which states that $G(U)\leq \Biggl[{(m-2)(m-n+1)\over n-1}\Biggr]-1$ if $$m$$ is large enough. A large number of results are also discussed.

##### MSC:
 11B75 Other combinatorial number theory 11P70 Inverse problems of additive number theory, including sumsets 20K99 Abelian groups 05D05 Extremal set theory
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