A Geometric Perspective on the MSTD Question

@article{Miller2019AGP,
  title={A Geometric Perspective on the MSTD Question},
  author={Steven J. Miller and Carsten Peterson},
  journal={Discrete \& Computational Geometry},
  year={2019},
  pages={1-24}
}
A more sums than differences (MSTD) set A is a subset of $$\mathbb {Z}$$Z for which $$|A+A| > |A-A|$$|A+A|>|A-A|. Martin and O’Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of $$\{1, \dots , n\}$${1,⋯,n} are MSTD as $$n \rightarrow \infty $$n→∞. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, $$\mathbb {I}$$I, and explore the MSTD question for such… 
2 Citations
Distribution of Missing Differences in Diffsets
Lazarev, Miller and O'Bryant investigated the distribution of $|S+S|$ for $S$ chosen uniformly at random from $\{0, 1, \dots, n-1\}$, and proved the existence of a divot at missing 7 sums (the
The Bidirectional Ballot Polytope
TLDR
It is proved that every $(2n-1)$-dimensional unit cube can be partitioned into isometric copies of the $\Theta(2^n/n)$, which forms a convex polytope sitting inside the unit cube, which is referred to as the bidirectional ballotpolytope.

References

SHOWING 1-10 OF 33 REFERENCES
Sets characterized by missing sums and differences
Some explicit constructions of sets with more sums than differences
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
Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs
TLDR
The main result is that g_m(t) is a quasipolynomial in t which satisfies a combinatorial reciprocity theorem and an analogue of Stanley's theorem to mixed graphs, which connects their chromatic polynomials to acyclic orientations.
Problems in additive number theory, I
Talk at the Atelier en combinatoire additive (Workshop on Arithmetic Combinatorics) at the Centre de recherches mathématiques at the Université de Montréal on April 8, 2006. Definition 1. A problem
Many sets have more sums than differences
Since addition is commutative but subtraction is not, the sumset S+S of a finite set S is predisposed to be smaller than the difference set S-S. In this paper, however, we show that each of the three
When almost all sets are difference dominated
TLDR
The heart of the approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p, and exhibits a threshold phenomenon regarding the ratio of the size of the difference- to the sumset.
SETS WITH MORE SUMS THAN DIFFERENCES
Let A be a finite subset of the integers or, more generally, of any abelian group, written additively. The set A has more sums than di! erences if |A + A| > |A ! A|. A set with this property is
On A Conjecture of Conway
  • J. Marica
  • Mathematics
    Canadian Mathematical Bulletin
  • 1969
Problem 7 of Section VI of H. T. Croft's "Research Problems" (August, 1967 edition) is by J. H. Conway: A is a finite set of integers {ai}. A + A denotes {ai + aj}, A - A denotes {ai - aj}. Prove
...
...