Restricted-sum-dominant sets

TL;DR

This paper investigates restricted sum-dominant sets in additive groups, proving their infinite existence for many sizes and providing explicit constructions of such sets.

Contribution

It establishes the infinite existence of restricted sum-dominant sets of various sizes and offers explicit constructions.

Findings

Infinitely many RSD sets exist for infinitely many sizes.

Explicit constructions of RSD sets are provided.

RSD sets are shown to be abundant in the integers.

Abstract

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Define $A + A := \{a + b : a, b \in A\}$ and $A \dotplus A := \{a + b : a, b \in A~\text{and}~ a \neq b\}$. The set $A$ is called a {\em sum-dominant (SD) set} if $|A + A| > |A - A|$, and it is called a {\em restricted sum-domonant (RSD) set} if $|A \dotplus A| > |A - A|$. In this paper, we prove that for infinitely many positive integers $k$, there are infinitely many RSD sets of integers of cardinality $k$. We also provide an explicit construction of infinite sequence of RSD sets.