A Note on Additive Complements

TL;DR

This paper extends previous work on additive complements, showing the existence of such sequences with specific asymptotic behaviors in their counting functions, including a limit superior of 2 and a difference of 1 infinitely often.

Contribution

It constructs additive complements with particular asymptotic properties, advancing understanding of their possible growth and sumset characteristics.

Findings

Existence of additive complements with limsup of A(x)B(x)/x equal to 2

Construction of additive complements where A(x)B(x) - x = 1 infinitely often

Extension of Liu and Fang's results on additive complements

Abstract

Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of A and B. In this paper, we extend the results of Liu and Fang in 2016 and obtain some results on additive complements. For example, we prove that there exist additive complements $A$ and $B$ such that $\limsup_{x\to+\infty} A(x)B(x)/x= 2$ and $A(x)B(x) - x = 1$ for infinitely positive integers $x$.