加法补集的Narkiewicz条件  

Narkiewicz's condition on additive complements

作　　者：陈永高[1] 方金辉 Yong-Gao Chen;Jin-Hui Fang

机构地区：[1]南京师范大学数学科学学院,南京210023

出　　处：《中国科学：数学》2024年第9期1207-1218,共12页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12171243和12171246)资助项目。

摘　　要：对于无穷非负整数集合A和B,若所有充分大的整数n均可以表示为a+b,其中a∈A,b∈B则称A,B为加法补集.用A(x)和B(x)分别表示A和B的计数函数.Narkiewicz(1960)证明了:对于加法补集A,B,若lim_(x→∞)A(x)B(x)/x=1,则lim_(x→∞)A(2x)/A(x)=1,或者A由B替代.本文给出此结果的简化证明,并推广了此结果.另外,本文还证明了,设a和b是满足1 a 2和a b的任意实数,则存在正整数集合A使得lim inf_(x→∞)A(2x)/A(x)=a且lim sup_(x→∞)A(2x)/A(x)=b.Two infinite sets A;B of nonnegative integers are called additive complements,if all sufficiently large integers could be expressed as a+b,where a 2 A,b 2 B.Narkiewicz(1960)proved that:for any additive complements A;B,if lim_(x→∞)A(x)B(x)=x=1,then lim_(x→∞)A(2x)=A(x)=1,or this holds with A replaced by B.In this paper,we give a simplified proof for this result and generalize it.Furthermore,we also prove that:for any real numbers a;b with 16 a 62 and a 6 b,there exists a set A of positive integers such that lim inf_(x→∞)A(2x)=A(x)=a and lim sup_(x→∞)A(2x)=A(x)=b.

关 键 词：加法补集 Narkiewicz条件 上极限 

分 类 号：O144[理学—数学]