Two addition theorems. (English) Zbl 0189.29701
Summary: The following theorems are proved:
(1) Let \(A\oplus B=A\cup B\cup (A+B)\). If \(G\) is a finite Abelian group and \(A_1+\ldots+A_k\) subsets of \(G\) with \(|A_1|+\ldots+|A_k|\geq |G|\) then either \(A_1\oplus \ldots\oplus A_k=G\) or \(0\in A_2+\ldots+A_k\). For \(k=2\) this statement is true for any group.
(2) Let \(a_1, \ldots, a_{p+k-1}\) be a sequence of \(p+k-1\) integers. Then it is possible to select \(k\) distinct indices \(i_1,\ldots, i_k\) such that \(a_{i_1}+\ldots+a_{i_k}\equiv 0\pmod p\).
By means of (2), the proof of a theorem of [P. Erdős, A. Ginzburg and A. Ziv [Bull. Res. Council Israel, Sect. F 10, 41–43 (1961; Zbl 0063.00009)] can be considerably simplified.
(1) Let \(A\oplus B=A\cup B\cup (A+B)\). If \(G\) is a finite Abelian group and \(A_1+\ldots+A_k\) subsets of \(G\) with \(|A_1|+\ldots+|A_k|\geq |G|\) then either \(A_1\oplus \ldots\oplus A_k=G\) or \(0\in A_2+\ldots+A_k\). For \(k=2\) this statement is true for any group.
(2) Let \(a_1, \ldots, a_{p+k-1}\) be a sequence of \(p+k-1\) integers. Then it is possible to select \(k\) distinct indices \(i_1,\ldots, i_k\) such that \(a_{i_1}+\ldots+a_{i_k}\equiv 0\pmod p\).
By means of (2), the proof of a theorem of [P. Erdős, A. Ginzburg and A. Ziv [Bull. Res. Council Israel, Sect. F 10, 41–43 (1961; Zbl 0063.00009)] can be considerably simplified.
MSC:
| 11B75 | Other combinatorial number theory |
