Generalized More Sums Than Differences Sets
Geoffrey Iyer, Oleg Lazarev, Steven J. Miller, Liyang Zhang

TL;DR
This paper generalizes the concept of sum-dominant sets to multiple sums and differences, proving positive percentages of such sets exist, developing explicit constructions, and analyzing their limiting behaviors and generational properties.
Contribution
It extends previous work by generalizing sum-dominant sets to multiple sums and differences, providing explicit constructions, and analyzing their asymptotic and generational behaviors.
Findings
A positive percentage of sets are sum-dominant for all nontrivial sum/difference combinations.
Explicit constructions of such sets are provided.
Most sets are not k-generational for all k, despite positive percentages for each fixed k.
Abstract
A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set such that . Though it was believed that the percentage of subsets of that are sum-dominant tends to zero, in 2006 Martin and O'Bryant \cite{MO} proved a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We prove that a positive percent of the time for all nontrivial choices of . Previous approaches proved the existence of infinitely many such sets given the existence of one; however, no method existed to construct such a set. We develop a new, explicit construction for one such set, and then extend to a positive percentage of sets. We extend these results further, finding sets that exhibit…
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Taxonomy
TopicsGame Theory and Voting Systems · Economic theories and models · Advanced Graph Theory Research
