Asymptotic complements in the integers

JOURNAL OF NUMBER THEORY

卷:213

Article

Biswas, Arindam^1,2  Saha, Jyoti Prakash³ 

[1] Univ Wien, Fak Math, Oskar Morgenstern Pl 1, A-1090 Vienna, Austria

[2] Erwin Schrodinger Int Inst Math & Phys ESI, Boltzmanngasse 9, A-1090 Vienna, Austria

[3] Indian Inst Sci Educ & Res Bhopal, Dept Math, Bhopal Bypass Rd, Bhopal 462066, Madhya Pradesh, India

关键词: Sumsets;    Additive complements;    Asymptotic complements;    Minimal complements;    Additive number theory;   

DOI  :  10.1016/j.jnt.2019.11.015

来源: Elsevier

【 摘 要 】

Let Wbe a nonempty subset of the set of integers Z. A nonempty subset Cof Zis said to be an asymptotic complement to W if W + C contains almost all the integers except a set of finite size. The set C is said to be a minimal asymptotic complement to Wif Cis an asymptotic complement to W, but C\{c} is not an asymptotic complement to Wfor every c is an element of C. Asymptotic complements have been studied in the context of representations of integers since the time of Erdos, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathanson. In this article, we study minimal asymptotic complements in Z and deal with a problem of Nathanson on their existence and their inexistence. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free