A generalization of sumset and its applications

Abstract

Listen

Translate article

Let A be a nonempty finite subset of an additive abelian group G and let r and h be positive integers. The generalized h-fold sumset of A, denoted by $$h^{(r)}A$$ , is the set of all sums of h elements of A, where each element appears in a sum at most r times. The direct problem for $$h^{(r)}A$$ is to find a lower bound for $$|h^{(r)}A|$$ in terms of |A|. The inverse problem for $$h^{(r)}A$$ is to determine the structure of the finite set A for which $$|h^{(r)}A|$$ is minimal with respect to some fixed value of |A|. If $$G = \mathbb {Z}$$ , the direct and inverse problems are well studied. In case of $$G = \mathbb {Z}/p\mathbb {Z}$$ , p a prime, the direct problem has been studied very recently by Monopoli (J. Number Theory, 157 (2015) 271–279). In this paper, we express the generalized sumset $$h^{(r)}A$$ in terms of the regular and restricted sumsets. As an application of this result, we give a new proof of the theorem of Monopoli and as the second application, we present new proofs of direct and inverse theorems for the case $$G = \mathbb {Z}$$ .