Two addition theorems

AbstractThe following theorems are proved:o(1)Let A⊕B=A∪B∪(A+B). If G is a finite Abelian group and A1+…+Ak subsets of G with |A1|+…+|Ak|≥|G| then either A1⊕…⊕Ak=G or 0∈A2+…+Ak. For k=2 this statement is true for any group.(3)Let a1, …, ap+k−1 be a sequence of p+k−1 integers. Then it is possible to select k distinct indices i1, …, ik such that ai1+…+aik≡0(mod p).By means of (2), the proof of a theorem of Erdös, Ginzburg, and Ziv can be considerably simplified