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arXiv · 2306.16025

The lower bound of weighted representation function

Shi-Qiang Chen
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Abstract

For any given set $A$ of nonnegative integers and for any given two positive integers $k_1,k_2$, $R_{k_1,k_2}(A,n)$ is defined as the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, we prove that if integer $k\geq2$ and set $A\subseteq\mathbb{N}$ such that $R_{1,k}(A,n)=R_{1,k}(\mathbb{N}\setminus A,n)$ holds for all integers $n\geq n_0$, then $R_{1,k}(A,n)\gg \log n$.

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Shi-Qiang Chen. 2023-06-28. The lower bound of weighted representation function. https://arxiv.org/abs/2306.16025

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