On almost perfect additive complements

Abstract

Let $A, B$ be infinite sets of non-negative integers with $n\in A+B$ for all sufficiently large integers $n$. We prove that if $A(x)B(x)-x$ does not tend to infinity, then $\limsup_{x\to\infty}\frac{A(x)B(x)}{x}\in\Big[\frac{3}{2}, 2\Big].$ The lower bound is sharp, and confirms a conjecture of Chen. Our proof is based on the weak limits and the convolution factorization theorem of Gabardo and Lai. A discrete proof of the same bound was also given by Zhang. Motivated by the known spectrum of perfect additive complements, we conjecture that the interval $(3/2, 8/5)$ is also excluded for almost perfect additive complements.