2024/10/30 by Bhuwanesh Rao Patil, Patil, Bhuwanesh Rao, Mohan
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical and Theoretical Analysis #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2410.22664
openalex publication_date 2024/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∈ A and y∈ B. Erdős 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=\ai: i∈ ℕ\ is a set of natural numbers such that ai1, then there exists a set B⊂ ℕ∖ A such that B is a sparse additive complement of the set A.