Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.2

When Sets Can and Cannot Have Sum-Dominant Subsets

Hùng Việt Chu
Department of Mathematics
Washington and Lee University
Lexington, VA 24450

Nathan McNew
Department of Mathematics
Towson University
Towson, MD 21252

Steven J. Miller
Department of Mathematics
Williams College
Williamstown, MA 01267

Victor Xu and Sean Zhang
Department of Mathematics
Carnegie Mellon University
Pittsburgh, PA 15213


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

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Received June 1 2018; revised versions received August 21 2018; November 13 2018; November 16 2018. Published in Journal of Integer Sequences, November 23 2018.

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