On sets with sum and difference structure
Jin-Hui Fang, Csaba S\'andor

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 of nonnegative integers and an integer , let be the number of representations of as and be the number of representations of as , where . In this paper, we determine the sets such that for every nonnegative integer . We also consider the \emph{difference} structure and prove that: there exist sets and of nonnegative integers such that for all large , and for any given nonnegative integer , we have for infinitely many positive integers . Other related results are also contained.
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Taxonomy
Topicsgraph theory and CDMA systems · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
