Minimal complementary sets

Abstract:

Let G be a group on which a measure m is defined. If $A,B \subset G$ we define $A \oplus B = C = \{ c|c = a + b,a \in A,b \in B\}$. By ${A_k} \subset G$ we denote a subset of G consisting of k elements. Given ${A_k}$ we define $s({A_k}) = \inf m\{ B|B \subset G,{A_k} \oplus B = G\}$ and ${c_k} = {\sup _{{A_k} \subset G}}s({A_k})$. Theorems 1, 2, and 3 deal with the problem of determining ${c_k}$. In the dual problem we are given B, $m(B) > 0$, and required to find minimal A such that $A \oplus B = G$ or, sometimes, $m(A \oplus B) = m(G)$. Theorems 5 and 6 deal with this problem.