关于加法补集的一个注记(英文)

A Note on Additive Complements

设A和B是正整数集合,若它们的和A+B=a+b:a∈A,b∈B}包含所有充分大的整数,则称A,B为加法补集.本文推广了陈永高和方金辉关于加法补集的一个结论,特别证明了:存在加法补集A,B满足lim sup_(x→∞)(A(x)B(x))/x=(22)/(13),而且存在无穷多个正整数x使得A(x)B(x)-x=1.

We call two sets A and B of non-negative integers additive complements if their sum contains all sufficiently large integers. Let A（x） and B（x） be the counting functions of A and B, respectively. In this paper, we extend the result in [Proc. Amer. Math. Soc., 2011, 139（3）： 881-883]. In particular, we prove that there exist additive complements A and B such that limsupx→∞[A（x）B（x）]/z=22/13 and A（x）B（x） - x = 1 for infinitely many positive integers x.