On function $SX$ of additive complements

TL;DR

This paper investigates the function SX related to perfect additive complements of nonnegative integers, determining its exact value and bounds, thus advancing understanding of the structure of such complements.

Contribution

The paper precisely determines the value of SX for perfect additive complements and establishes the absolute lower bound for SX among additive complements.

Findings

Calculated the exact value of SX for perfect additive complements.

Established the absolute lower bound of SX for additive complements.

Fixed the infimum of SX for perfect additive complements.

Abstract

Two sets $A,B$ of nonnegative integers are called \emph{additive complements}, if all sufficiently large integers can be expressed as the sum of two elements from $A$ and $B$. We further call $A,B$ \emph{perfect additive complements} if every nonnegative integer can be uniquely expressed as the sum of two elements from $A$ and $B$. Let $A(x)$ be the counting function of $A$. In this paper, we focus on the function $SX$, where $SX=\limsup_{x\rightarrow\infty}\frac{\max\{A(x),B(x)\}}{\sqrt{x}}$ was introduced by Erd\H{o}s and Freud in 1984. As a main result, we determine the value of $SX$ for perfect additive complements and further fix the infimum. We also give the absolute lower bound of $SX$ for additive complements.