A generalization of Kneser's addition theorem
For an additive Abelian group \(G\) let \(A,B\subseteq G\) and \(g\in G\). The sumset of \(A\) and \(B\) is the set \(A+B :=\{a+b: a\in A, b\in B\}\) and \(g+A= A+g:= \{g\}+A\). Any set of the form \(g+A\) is called a shift of \(A\). The stabilizer of \(A\) is defined by \(\text{stab}(A):=\{g\in G: g+A= A\}\). In this note the principal subject matter is sequences of sets. If \({\mathbf A}= (A_1,A_2,\dots, A_m)\) is a sequence of finite subsets of \(G\) and \(\ell\leq m\), then the authors define the set \[ \Sigma^\ell({\mathbf A}):= \{a_{i_1}+\cdots+ a_{i_ell}: 1\leq i_1<\cdots< i_\ell\leq m\;\text{and}\;a_{i_j}\in A_{i_j}\;\text{for every}\;1\leq j\leq\ell\}. \] So \(\Sigma^\ell({\mathbf A})\) is the set of all elements of \(G\) which can be represented as a sum of terms from distinct members of \({\mathbf A}\). The main result of this note is the following Theorem giving a lower bound for \(|\Sigma^\ell({\mathbf A})|\): Let \({\mathbf A}= (A_1,\dots, A_m)\) be a sequence of finite subsets of \(G\), let \(\ell\leq m\) and \(H= \text{stab}(\Sigma^\ell({\mathbf A}))\). If \(\Sigma^\ell({\mathbf A})\) is nonempty, then \[ \left|\Sigma^\ell({\mathbf A})\right|\geq|H|\left(1- \ell+ \sum_{Q\in G/H}\min\{\ell,|\{i\in \{1,\dots, m\}: A_i\cap Q\neq\emptyset\}|\}\right). \] For the special case \(m=\ell=2\) this result is equivalent to a theorem of \textit{M. Kneser} [Math. Z. 58, 459--484 (1953; Zbl 0051.28104)] and the arguments used here give a new proof of this theorem. Further some recent results of other authors are extended in this paper.
scientific article; zbMATH DE number 5530599
scientific article