Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
-
Steven J. Miller
,
Luc Robinson
Abstract.
A More Sums Than Differences (MSTD) set is a set of integers
whose sumset 
is larger than its difference set 
. While it is known that as 
a positive percentage of subsets of 
are MSTD sets, the methods to prove this are probabilistic and do not yield nice, explicit constructions. Recently Miller, Orosz and Scheinerman (2010) gave explicit constructions of a large family of MSTD sets; though their density is less than a positive percentage, their family's density among subsets of is at least 
for some 
, significantly larger than the previous constructions, which were on the order of 
. We generalize their method and explicitly construct a large family of sets A with 
. The additional sums and differences allow us greater freedom than by Miller, Orosz and Scheinerman (2010), and we find that for any 
the density of such sets is at least 
. In the course of constructing such sets we find that for any integer k there is an A such that 
, and show that the minimum span of such a set is 30.
© 2012 by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Masthead
- Two-Color Babylon
- On Abelian and Additive Complexity in Infinite Words
- Approximations of Additive Squares in Infinite Words
- The Googol-th Bit of the Erdős–Borwein Constant
- Odd Repdigits to Small Bases Are Not Perfect
- Enumeration of the Degree Sequences of Line-Hamiltonian Multigraphs
- The Characteristic Sequence and p-Orderings of the Set of d-th Powers of Integers
- Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}
- Sum-Products Estimates with Several Sets and Applications
- The 2-Adic, Binary and Decimal Periods of 1/3k Approach Full Complexity for Increasing k
- A Proof of Catalan's Convolution Formula
- Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
- The Impossibility of Certain Types of Carmichael Numbers
- Integer Subsets with High Volume and Low Perimeter
- Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables
- A Generalized Diagonal Wythoff Nim
- A Recursive Process Related to a Partizan Variation of Wythoff
- The (r1,...,rp)-Stirling Numbers of the Second Kind
- Factor Frequencies in Languages Invariant Under Symmetries Preserving Factor Frequencies
- On k-Lehmer Numbers
Articles in the same Issue
- Masthead
- Two-Color Babylon
- On Abelian and Additive Complexity in Infinite Words
- Approximations of Additive Squares in Infinite Words
- The Googol-th Bit of the Erdős–Borwein Constant
- Odd Repdigits to Small Bases Are Not Perfect
- Enumeration of the Degree Sequences of Line-Hamiltonian Multigraphs
- The Characteristic Sequence and p-Orderings of the Set of d-th Powers of Integers
- Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}
- Sum-Products Estimates with Several Sets and Applications
- The 2-Adic, Binary and Decimal Periods of 1/3k Approach Full Complexity for Increasing k
- A Proof of Catalan's Convolution Formula
- Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets
- The Impossibility of Certain Types of Carmichael Numbers
- Integer Subsets with High Volume and Low Perimeter
- Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables
- A Generalized Diagonal Wythoff Nim
- A Recursive Process Related to a Partizan Variation of Wythoff
- The (r1,...,rp)-Stirling Numbers of the Second Kind
- Factor Frequencies in Languages Invariant Under Symmetries Preserving Factor Frequencies
- On k-Lehmer Numbers
