On a problem of Chen and Fang related to infinite additive complements

TL;DR

This paper extends Danzer's construction of infinite additive complements of nonnegative integers, addressing a problem posed by Chen and Fang and advancing understanding of their structure.

Contribution

It provides an extended construction of infinite additive complements, solving a specific open problem posed by Chen and Fang.

Findings

Constructed infinite additive complements with specific density properties

Extended Danzer's method to solve Chen and Fang's problem

Achieved a new understanding of the structure of additive complements

Abstract

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.