Complementary sets of finite sets
Given an additive group \(G\) and \({\mathcal A}\subset G\), a set \({\mathcal S}\subset G\) is called complementary to \(\mathcal A\) in \(G\) if \({\mathcal A}+{\mathcal S}=\{a+s:a\in{\mathcal A}, s\in{\mathcal S}\}=G\). In the paper the groups \(\mathbb Z\), \({\mathbb Z}/(m{\mathbb Z})\), \(\mathbb R\), and \({\mathbb R}/{\mathbb Z}\) are considered, and some sufficient conditions for the existence of a complementary set \(\mathcal S\) with ``density'' not much larger than \(1/|{\mathcal A}|\) are given, where \(\mathcal A\) is finite. Some structural descriptions of complementary sets of small density which depend on approximation properties of elements of \(\mathcal A\) are also given.
scientific article; zbMATH DE number 1942706