Correct order on some certain weighted representation functions

2023/06/29 by Shi--Qiang Chen, Chen, Shi--Qiang, Yuchen Ding +4
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2306.16820

openalex publication_date 2023/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ℕ be the set of all nonnegative integers. For any positive integer k and any subset A of nonnegative integers, let r_1,k(A,n) be the number of solutions (a₁,a₂) to the equation n=a₁+ka₂. In 2016, Qu proved that \liminf_n→∞r_1,k(A,n)=∞ providing that r_1,k(A,n)=r_1,k(ℕ∖ A,n) for all sufficiently large integers, which answered affirmatively a 2012 problem of Yang and Chen. In a very recent article, another Chen (the first named author) slightly improved Qu's result and obtained that \liminf_n→∞\fracr_1,k(A,n)log ngt;0. In this note, we further improve the lower bound on r_1,k(A,n) by showing that \liminf_n→∞\fracr_1,k(A,n)ngt;0. Our bound reflects the correct order of magnitude of the representation function r_1,k(A,n) under the above restrictions due to the trivial fact that r_1,k(A,n)≤ n/k.

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