A Note on Additive Complements

2022/05/09 by Fang-Yu Ma, Ma, Fang-Yu
Mathematics · #Advanced Topology and Set Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2205.04128

openalex publication_date 2022/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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→+∞ A(x)B(x)/x= 2 and A(x)B(x) - x = 1 for infinitely positive integers x.

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