Identical representation functions of linear forms

TL;DR

This paper extends Nathanson's classical results on representation functions of sums of two numbers to more general linear forms, exploring conditions for when different sets have identical representation functions.

Contribution

It introduces new conditions for sets to have identical representation functions for general linear forms, broadening the scope beyond sums of two numbers.

Findings

Extended Nathanson's results to linear forms

Identified conditions for identical representation functions

Explored related problems in additive number theory

Abstract

For a set of natural numbers $A$, let $R_{A}(n)$ be the number of representations of a natural number $n$ as the sum of two terms from $A$. Many years ago, Nathanson studied the conditions for the set $A$ and $B$ of natural numbers that are needed to guarantee that $R_{A}(n) = R_{B}(n)$ for every positive integer $n$. In the last decades, similar questions have been studied by many authors. In this paper, we extend Nathanson's result to representation functions associated to linear forms and we study related problems.