zbMATH — the first resource for mathematics

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.

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