On sets with sum and difference structure

TL;DR

This paper characterizes sets of nonnegative integers with unique sum representations and explores their difference structures, revealing diverse representation behaviors and asymptotic properties.

Contribution

It determines sets with unique sum representations and constructs sets with prescribed difference representation patterns, advancing understanding of additive and subtractive set structures.

Findings

Sets with unique sum representations are characterized.

Existence of sets with bounded difference representations for infinitely many integers.

Construction of sets with specific asymptotic and difference representation properties.

Abstract

For nonempty sets $A,B$ of nonnegative integers and an integer $n$, let $r_{A,B}(n)$ be the number of representations of $n$ as $a+b$ and $d_{A,B}(n)$ be the number of representations of $n$ as $a-b$, where $a\in A, b\in B$. In this paper, we determine the sets $A,B$ such that $r_{A,B}(n)=1$ for every nonnegative integer $n$. We also consider the \emph{difference} structure and prove that: there exist sets $A$ and $B$ of nonnegative integers such that $r_{A,B}(n)\ge 1$ for all large $n$, $A(x)B(x)=(1+o(1))x$ and for any given nonnegative integer $c$, we have $d_{A,B}(n)=c$ for infinitely many positive integers $n$. Other related results are also contained.