The lower bound of weighted representation function

TL;DR

This paper establishes a lower bound of logarithmic growth for the weighted representation function under certain symmetric conditions, revealing fundamental limits of such representations in number theory.

Contribution

It proves that if a set's weighted representation function matches that of its complement beyond a certain point, then it must grow at least logarithmically.

Findings

The weighted representation function grows at least as fast as a constant times log n.

Symmetry condition implies a logarithmic lower bound on the representation function.

Provides insight into the structure of sets with equal representation functions.

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 $R_{1,k}(A,n)\gg \log n$.