On the monotonicity of additive representation functions

TL;DR

This paper investigates the monotonicity properties of three variants of additive representation functions, resolving an open question, partially answering another, and providing a simpler proof for a known result.

Contribution

It advances understanding of additive representation functions by clarifying their monotonicity behavior and simplifying existing proofs.

Findings

Resolved an open question on monotonicity

Partially answered a second open question

Provided a simpler proof for a known result

Abstract

We study the monotonicity behavior of three slightly differently defined additive representation functions (as initiated by Erd\H{o}s, S\'{a}rk\"{o}zy and S\'{o}s), answering one open question and another one partially, and give a slightly simpler proof for a result of Chen and Tang.