Some explicit constructions of sets with more sums than differences

By Peter V. Hegarty. 2006

58 citations recorded by the source corpus.

We present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a unique MSTD subset of {\bf Z} of size 8. Secondly, starting from some examples of size 9, we present several new constructions of infinite families of MSTD sets. Thirdly we show that for every fixed ordered pair of non-negative integers (j,k), as n -> \infty a positive proportion of the subsets of {0,1,2,...,n} satisfy |A+A| = (2n+1) - j, |A-A| = (2n+1) - 2k.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/2d976fb7f6b543ad8779b267da83810e4782c033

How researchers see this paper

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In Algebra and Number Theory

1. Yufei Zhao
2. Melvyn B. Nathanson
3. Nathan Kaplan

In Discrete Mathematics and Combinatorics

1. Yufei Zhao
2. Melvyn B. Nathanson
3. Peter V. Hegarty
4. Nathan Kaplan

In Electrical and Electronic Engineering

1. Yufei Zhao
2. Melvyn B. Nathanson
3. Peter V. Hegarty

Citations in the corpus

• Sets characterized by missing sums and differences (2009, background): “Hegarty [1] showed that, for d even, the limit ρs,d is positive provided that it exists.”
• SETS WITH MORE SUMS THAN DIFFERENCES (2006, background): “Hegarty [2] proved that there is no MSTD set of cardinality less than 8, and that the set A1 is, up to affine equivalence, the unique MSTD set of cardinality 8.”