On additive complements. III

TextTwo infinite sequences A and B of non-negative integers are called additive complements  , if their sum contains all sufficiently large integers. Let A(x)A(x) and B(x)B(x) be the counting functions of A and B. In 1994, Sárközy and Szemerédi proved that, for additive complements A and B  , if limsupA(x)B(x)/x⩽1, then A(x)B(x)−x→+∞A(x)B(x)−x→+∞ as x→+∞x→+∞. In 2010, the authors generalized this result and proved that if limsupA(x)B(x)/x<5/4 or limsupA(x)B(x)/x>2, then A(x)B(x)−x→+∞A(x)B(x)−x→+∞ as x→+∞x→+∞. In 2011, the authors pointed out that the constant 2 cannot be improved. In this paper, we improve 5/4 to 3−3.VideoFor a video summary of this paper, please click here or visit http://youtu.be/fSTUWhPJdtw.