A generalization of sumsets of set of integers

作者：Mistri Raj Kumar; Pandey Ram Krishna^*

来源：Journal of Number Theory, 2014, 143: 334-356.

DOI：10.1016/j.jnt.2014.04.005

• Arithmetic progression
• h-Fold sumsets
• Direct and inverse problems

Let A be a nonempty finite set of integers. The h-fold sumset of A, denoted by hA, is the set of all sums of h elements of A with repetitions allowed. A restricted h-fold sumset of A, denoted by h boolean AND A, is the set of all sums of h distinct elements of A. For h %26gt;= 1 and r %26gt;= 1, we define a generalized h-fold sumset, denoted by h((r))A, which is the set of all sums of h elements of A, where each element appearing in the sum can be repeated at most r times. Thus the h-fold sumset hA and the restricted h-fold sumset h boolean AND A are particular cases of the sumset h((r))A for r = h and r = 1, respectively. The direct problem for h((r))A is to find a %26apos;lower bound for vertical bar h((r))A vertical bar in terms of vertical bar A vertical bar. The inverse problem for h((r))A is to determine the structure of the finite set A of integers for which vertical bar h((r))A vertical bar is minimal. In this paper we solve both the problems.

• 出版日期2014-10