On additive complements in natural numbers

Abstract

We prove that for additive complements $A$ and $B$, if $$\underset{x\to +\infty }{\mathrm{limsup}}\frac{A(x)B(x)}{x}<\frac{4}{3},$$ then $$A(x)B(x)-x\to +\infty (x\to +\infty ).$$ This improves the earlier upper bound $3-\sqrt{3}$ due to Fang and Chen.