An additive theorem and restricted sumsets

Abstract

Let $G$ be any additive abelian group with cyclic torsion subgroup, and let $A$, $B$ and $C$ be finite subsets of $G$ with cardinality $n>0$. We show that there is a numbering $\{a_i\}_{i=1}^n$ of the elements of $A$, a numbering $\{b_i\}_{i=1}^n$ of the elements of $B$ and a numbering $\{c_i\}_{i=1}^n$ of the elements of $C$, such that all the sums $a_i+b_i+c_i (1\ls i\ls n)$ are (pairwise) distinct. Consequently, each subcube of the Latin cube formed by the Cayley addition table of $\Z/N\Z$ contains a Latin transversal. This additive theorem is an essential result which can be further extended via restricted sumsets in a field.