Constructing MSTD Sets Using Bidirectional Ballot Sequences
نویسنده
چکیده
A more sums than differences (MSTD) set is a finite subset S of the integers such that |S + S| > |S − S|. We construct a new dense family of MSTD subsets of {0, 1, 2, . . . , n − 1}. Our construction gives Θ(2n/n) MSTD sets, improving the previous best construction with Ω(2/n) MSTD sets by Miller, Orosz, and Scheinerman.
منابع مشابه
Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum
Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the e...
متن کاملFinding and Counting MSTD sets
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 constructi...
متن کاملFringe Pairs in Generalized Mstd Sets
A More Sums Than Differences (MSTD) set is a set A for which |A+A| > |A−A|. Martin and O’Bryant proved that the proportion of MSTD sets in {0, 1, . . . , n} is bounded below by a positive number as n goes to infinity. Iyer, Lazarev, Miller and Zhang introduced the notion of a generalized MSTD set, a set A for which |sA− dA| > |σA − δA| for a prescribed s + d = σ + δ. We offer efficient construc...
متن کاملExplicit Constructions of Infinite Families of Mstd Sets
We explicitly construct infinite families of MSTD (more sums than differences) sets, i.e., sets where |A+A| > |A−A|. There are enough of these sets to prove that there exists a constant C such that at least C/r of the 2 subsets of {1, . . . , r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/2 for some polynomial f(r)). We co...
متن کاملCounting MSTD Sets in Finite Abelian Groups
In an abelian group G, a more sums than differences (MSTD) set is a subset A ⊂ G such that |A+A| > |A−A|. We provide asymptotics for the number of MSTD sets in finite abelian groups, extending previous results of Nathanson. The proof contains an application of a recently resolved conjecture of Alon and Kahn on the number of independent sets in a regular graph.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009