A generalization of sumsets of set of integers

JOURNAL OF NUMBER THEORY

卷:143

Article

Mistri, Raj Kumar¹  Pandey, Ram Krishna² 

[1] Indian Inst Technol Patna, Dept Math, Patna 800013, Bihar, India

[2] Indian Inst Technol Roorkee, Dept Math, Roorkee 247667, Uttar Pradesh, India

关键词: Arithmetic progression;    h-Fold sumsets;    Direct and inverse problems;   

DOI  :  10.1016/j.jnt.2014.04.005

来源: Elsevier

【 摘 要 】

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 >= 1 and r >= 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 '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. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free