A note on the lower bound of representation functions

TL;DR

This paper investigates the lower bounds of representation functions for sets of nonnegative integers, showing that previous bounds are nearly optimal and exploring related properties of such sets.

Contribution

It demonstrates that the earlier established lower bounds are nearly optimal and presents additional results on the structure of representation functions.

Findings

The previous lower bound on R_2(A,n) is nearly best possible.

Conditions under which the representation functions are equal for large n.

Additional results on the behavior of representation functions for specific sets.

Abstract

For a set $A$ of nonnegative integers, let $R_2(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, $a<a'$. Let $A_0$ be the Thue-Morse sequence and $B_0=\mathbb{N}\setminus A_0$. Let $A\subset \mathbb{N}$ and $N$ be a positive integer such that $R_2(A,n)=R_2(\mathbb{N}\setminus A,n)$ for all $n\geq 2N-1$. Previously, the first author proved that if $|A\cap A_0|=+\infty$ and $|A\cap B_0|=+\infty$, then $R_2(A,n)\geq \frac{n+3}{56N-52}-1$ for all $n\geq 1$. In this paper, we prove that the above lower bound is nearly best possible. We also get some other results.