A generalization of sumsets of set of integers
Let \(A\) be a nonempty finite set of integers. Denote by \(k=|A|\) its cardinality. In additive number theory, for a given positive integer \(h\), one considers the following sumsets created from \(A\) : \(h A=\{x_1+\cdots+x_h:x_i\in A, 1\leq i\leq h\}\), and [for \(h\leq k\)] \(h^\land A=\{x_1+\cdots+x_h:x_i\in A, 1\leq i\leq h~,~x_i\neq x_j~\text{if}~i\neq j\}\). The best lower bounds for \(|hA|\) and \(|h^\land A|\) in terms of \(h\) and \(k,\) and the characterization of the sets \(A\) attaining these lower bounds are known. The reader can see the book by \textit{M. B. Nathanson} [Additive number theory. Inverse problems and the geometry of sumsets. New York, NY: Springer (1996; Zbl 0859.11003)] for formulation, references and proofs. The authors extend the above mentioned results by considering, for a given integer \(r\geq1,\) the sumset denoted by \(h^{(r)}A\) and defined as follows. Let \(A=\{a_0,a_1,\dots,a_{k-1}\}\). Then define \[ h^{(r)}A:=\{\sum_{i=0}^{k-1}r_ia_i~;~0\leq r_i\leq r~,~0\leq i\leq k-1~, \sum_{i=0}^{k-1}r_i=h\}~. \] The main theorem states: ``Let \(A\) be a nonempty set of \(k\) integers. Let \(r\) and \(h\) be integers such that \(1\leq r\leq h\leq rk.\) Set \(m:=\lfloor h/r\rfloor.\) Then \[ |h^{(r)}A|\geq mr(k-m)+(h-mr)(k-2m-1)+1~. \] This lower bound is best possible.'' The authors also find all sets for which equality in the preceding relation holds. The paper ends with two open problems. The first one is to generalize the above mentioned main theorem, replacing \(r\) by \(R:=(R_0,\dots,R_{k-1})\) and allowing in the sum each \(a_i\) to be repeated at most \(R_i\) times. This problem is now solved by \textit{Q.-H. Yang} and \textit{Y.-G. Chen} [``On the cardinality of general \(h\)-fold sumsets'', Eur. J. Comb. 47, 103--114 (2015; Zbl 1448.11025)].
scientific article; zbMATH DE number 6339013