On infinite additive complements

• 来源期刊: 中国科学（数学）
• 年，卷（期）: 2017,（10）
• 页码范围: 1779-1790
• 页数: 12页
• 语种: 中文
• DOI: 10.1007/s11425-016-0114-y

摘要

Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers.For a sequence T of non-negative integers,let T(x) be the number of terms of T not exceeding x.In 1994,Sárk(o)zy and Szemerédi confirmed a conjecture of Danzer by proving that,for infinite additive complements A and B,if lim sup A(x)B(x)/x ≤ 1,then A(x)B(x)-x → +∞ as x → +∞.In this paper,it is proved that,if A and B are infinite additive complements with limsup A(x)B(x)/x ＜ (4√2 + 2)/7 =1.093 …,then (A(x)B(x)-x)/min{A(x),B(x)} → +∞ as x → +∞.