Abstract
We review the basic theory of more sums than differences (MSTD) sets, specifically their existence, simple constructions of infinite families, the proof that a positive percentage of sets under the uniform binomial model are MSTD but not if the probability that each element is chosen tends to zero, and “explicit” constructions of large families of MSTD sets. We conclude with some new constructions and results of generalized MSTD sets, including among other items results on a positive percentage of sets having a given linear combination greater than another linear combination, and a proof that a positive percentage of sets are k-generational sum-dominant (meaning A, A + A, \(\ldots\), \(kA = A + \cdots + A\) are each sum-dominant).
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Notes
- 1.
Note that \(A + A + A - A = -(A - A - A - A)\); thus we might as well assume any linear combination has at least as many sums of A as differences of A.
- 2.
Requiring 0, 2n − 1 ∈ A is quite mild; we do this so that we know the first and last elements of A.
- 3.
As before, requiring 0, 2n − 1 ∈ A is quite mild and is done so that we know the first and last elements of A.
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Iyer, G., Lazarev, O., Miller, S.J., Zhang, L. (2014). Finding and Counting MSTD Sets. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory. Springer Proceedings in Mathematics & Statistics, vol 101. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1601-6_7
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