Sets characterized by missing sums and differences

@article{Zhao2011SetsCB,
  title={Sets characterized by missing sums and differences},
  author={Yufei Zhao},
  journal={Journal of Number Theory},
  year={2011},
  volume={131},
  pages={2107-2134}
}
  • Yufei Zhao
  • Published 12 November 2009
  • Mathematics
  • Journal of Number Theory

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References

SHOWING 1-10 OF 22 REFERENCES
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
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
Constructing MSTD Sets Using Bidirectional Ballot Sequences
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
Problems in additive number theory, III
Let ℕ, ℕ0, ℤ and ℕ d denote, respectively, the sets of positive integers, non-negative integers, integers and d-dimensional integral lattice points. Let G denote an arbitrary abelian group and let X
Constructing numerical semigroups of a given genus
Let ng denote the number of numerical semigroups of genus g. Bras-Amorós conjectured that ng possesses certain Fibonacci-like properties. Almost all previous attempts at proving this conjecture were
...
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