Generalized More Sums Than Differences sets
نویسندگان
چکیده
منابع مشابه
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 differences if |A + A| > |A − A|. A set with this property is called an MSTD set. This paper gives explicit constructions of families of MSTD sets of integers.
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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, socalled MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a unique MSTD subset of Z of size 8. Secondly, starting from some examples of size 9, we present several new constructions of infinite families of MSTD sets....
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A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set A ⊂ Z such that |A+A| < |A−A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O’Bryant [MO] proved that a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We p...
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Many fundamental questions in additive number theory (such as Goldbach’s conjecture, Fermat’s last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements contributes one sum and two differences, we expect that |A−A| > |A+A| for a finite set A. However, in 2006 Martin and O’Bryant showed that a positive proportion of su...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2012
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2011.10.006