Generalizations of a Curious Family of MSTD Sets Hidden By Interior Blocks

Abstract

A set $A$ is MSTD (more-sum-than-difference) or sum-dominant if $|A+A|>|A-A|$, and is RSD (restricted-sum dominant) if $|A\hat{+}A|>|A-A|$, where $A\hat{+}A$ is the set of sums of distinct elements in $A$. We study an interesting family of MSTD sets that have appeared many times in the literature (see the works of Hegarty, Martin and O'Bryant, and Penman and Wells). While these sets seem at first glance to be ad hoc, looking at them in the right way reveals a nice common structure. In particular, instead of viewing them as explicitly written sets, we write them in terms of differences between two consecutive numbers in increasing order. We denote this family by $\mathcal{F}$ and investigate many of its properties. Using $\mathcal{F}$, we are able to generate many sets $A$ with high value of $\log|A+A|/\log|A-A|$, construct sets $A$ with a fixed $|A+A|-|A-A|$ more economically than previous authors, and improve the lower bound on the proportion of RSD subsets of $\{0,1,2,\dots,n-1\}$ to about $10^{-25}$ (the previous best bound was $10^{-37}$). Lastly, by exhaustive computer search, we find six RSD sets with cardinality $15$, which is one lower than the smallest cardinality found to date, and find that $30$ is the smallest diameter of RSD sets.