When Sets Can and Cannot Have MSTD Subsets

Nathan McNew, Steven J. Miller, Victor Xu, Sean Zhang

Published 2016-08-10Version 1

A finite set of integers $A$ is a More Sums Than Differences (MSTD) set if $|A+A| > |A-A|$. While almost all subsets of $\{0, \dots, n\}$ are not MSTD, interestingly a small positive percentage are. We explore sufficient conditions on infinite sets of positive integers such that there are either no MSTD subsets, at most finitely many MSTD subsets, or infinitely many MSTD subsets. In particular, we prove no subset of the Fibonacci numbers is an MSTD set, establish conditions such that solutions to a recurrence relation have only finitely many MSTD subsets, and show there are infinitely many MSTD subsets of the primes.