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, S′ark¨ozy and Szemer′edi 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 lim sup A(x)B(x)/x<(4 2～(1/2) + 2)/7 = 1.093 …, then(A(x)B(x)-x)/min{A(x), B(x)} → +∞ as x → +∞.
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, Sarkozy and Szemer′edi 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 lim sup A（x）B（x）/x〈（√4 ＋ 2）/7 = 1.093 …, then（A（x）B（x）-x）/min{A（x）, B（x）} → ＋∞ as x → ＋∞.
基金
supported by National Natural Science Foundation of China (Grant Nos. 11671211 and 11371195)
the China Scholarship Council (Grant No. 201608320048)
the Priority Academic Program Development of Jiangsu Higher Education Institutions
