An additive theorem and restricted sumsets
Zhi‐Wei Sun · Mathematical Research Letters · 2008
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 } n i=1 of the elements of A, a numbering {b i } n i=1 of the elements of B and a numbering {c i } n i=1 of the elements of C, such that all the sums a i + b i + c i (1 i 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.