# Finding and Counting MSTD Sets

@article{Iyer2014FindingAC, title={Finding and Counting MSTD Sets}, author={Geoffrey Iyer and Oleg Lazarev and Steven J. Miller and Liyang Zhang}, journal={arXiv: Number Theory}, year={2014}, pages={79-98} }

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…

## 8 Citations

### Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets

- MathematicsIntegers
- 2012

In the course of constructing a large family of sets A, it is found that for any integer k there is an A such that, and it is shown that the minimum span of such a set is 30.

### Generalizations of a Curious Family of MSTD Sets Hidden By Interior Blocks

- Mathematics
- 2018

A set $A$ is MSTD (more-sum-than-difference) or sum-dominant if $|A+A|>|A-A|$ and is RSD (restricted-sum dominant) if $|A\hat{+}A|>|A-A|$, where $A\hat{+}A$ is the sumset of $A$ without a number…

### #A60 INTEGERS 19 (2019) INFINITE FAMILIES OF PARTITIONS INTO MSTD SUBSETS

- Mathematics
- 2019

A set A is MSTD (more-sum-than-di↵erence) if |A + A| > |A A|. Though MSTD sets are rare, Martin and O’Bryant proved that there exists a positive constant lower bound for the proportion of MSTD…

### Infinite Families of Partitions Into MSTD Subsets

- MathematicsIntegers
- 2019

An efficient method to partition $\{1,2,\ldots,r\}$ (for $r$ sufficiently large) into $k \ge 2$ MSTD subsets, positively answering a question raised by Asada et al. as to whether this is possible for all such $k$.

### Most Subsets Are Balanced in Finite Groups

- Mathematics
- 2014

The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach’s conjecture and Fermat’s last theorem) can be formulated in…

### Union of Two Arithmetic Progressions with the Same Common Difference Is Not Sum-dominant

- Mathematics
- 2019

Given a finite set $A\subseteq \mathbb{N}$, define the sum set $$A+A = \{a_i+a_j\mid a_i,a_j\in A\}$$ and the difference set $$A-A = \{a_i-a_j\mid a_i,a_j\in A\}.$$ The set $A$ is said to be…

### ON SETS WITH MORE PRODUCTS THAN QUOTIENTS

- Mathematics, Philosophy
- 2020

Given a finite set A ⊂ R\{0}, define A ·A = {ai · aj | ai, aj ∈ A}, A/A = {ai/aj | ai, aj ∈ A}, A+A = {ai + aj | ai, aj ∈ A}, A−A = {ai − aj | ai, aj ∈ A}. The set A is said to be MPTQ (more product…

### On sets with more products than quotients

- MathematicsRocky Mountain Journal of Mathematics
- 2020

Given a finite set $A\subset \mathbb{R}\backslash \{0\}$, define \begin{align*}&A\cdot A \ =\ \{a_i\cdot a_j\,|\, a_i,a_j\in A\},\\ &A/A \ =\ \{a_i/a_j\,|\,a_i,a_j\in A\},\\ &A + A \ =\ \{a_i +…

## References

SHOWING 1-10 OF 24 REFERENCES

### Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets

- MathematicsIntegers
- 2012

In the course of constructing a large family of sets A, it is found that for any integer k there is an A such that, and it is shown that the minimum span of such a set is 30.

### Some explicit constructions of sets with more sums than differences

- Mathematics
- 2007

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…

### SETS WITH MORE SUMS THAN DIFFERENCES

- Mathematics
- 2006

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…

### Many sets have more sums than differences

- Mathematics
- 2006

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

- MathematicsRandom Struct. Algorithms
- 2009

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.

### New bounds on nearly perfect matchings in hypergraphs: Higher codegrees do help

- MathematicsRandom Struct. Algorithms
- 2000

A general upper bound on U(H), based on the codegree sequence C2 (H), C3(H) is proved, which improves and generalizes many results on the topic, including those of Grable, AlonKim-Spencer, and Kostochka-Rödl.