  Two infinite sets $A$ and $B$ of nonnegative integers are called additive
complements if their sumset contains every nonnegative integer. In 1964, Danzer
constructed infinite additive complements $A$ and $B$ with $A(x)B(x) = (1 +
o(1))x$ as $x \rightarrow \infty$, where $A(x)$ and $B(x)$ denote the counting
function of the sets $A$ and $B$, respectively. In this paper we solve a
problem of Chen and Fang by extending the construction of Danzer.
