Correct order on some certain weighted representation functions

Abstract

Let $\mathbb{N}$ 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 $(a_1,a_2)$ to the equation $n=a_1+ka_2$. In 2016, Qu proved that $$\liminf_{n\rightarrow\infty}r_{1,k}(A,n)=\infty$$ providing that $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ 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 $$\liminf_{n\rightarrow\infty}\frac{r_{1,k}(A,n)}{\log n}>0.$$ In this note, we further improve the lower bound on $r_{1,k}(A,n)$ by showing that $$\liminf_{n\rightarrow\infty}\frac{r_{1,k}(A,n)}{n}>0.$$ 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)\le n/k.