On additive complements in the complement of a set of natural numbers

TL;DR

This paper investigates the existence of sparse additive complements within the complement of a set of natural numbers, providing a ratio test to determine when such complements exist based on the growth rate of the set.

Contribution

It introduces a ratio test criterion for the existence of sparse additive complements in the complement of a set of natural numbers, extending previous results on additive complements.

Findings

Existence of sparse additive complements under certain growth conditions

A ratio test criterion for verifying the existence of such complements

Proof that if the set's elements grow faster than a certain rate, complements exist

Abstract

Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. Erd\H{o}s proposed a conjecture that every infinite set of natural numbers has a sparse additive complement, and in 1954, Lorentz proved this conjecture. This article describes the existence or non-existence of those additive complements of the set $A$ that is a subset of the complement of $A$. We provide a ratio test to verify the existence of such additive complements. In precise, we prove that if $A=\{a_i: i\in \mathbb{N}\}$ is a set of natural numbers such that $a_i<a_{i+1} $ for $i \in \mathbb{N}$ and $\liminf_{n\rightarrow \infty } (a_{n+1}/a_{n})>1$, then there exists a set $B\subset \mathbb{N}\setminus A$ such that $B$ is a sparse additive complement of the set $A$.