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

Miller, Steven J.; Robinson, Luc; Pegado, Sean

doi:10.1515/integers-2012-0015

Cited by 9 publications

(7 citation statements)

References 12 publications

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“…Later, Miller et al [15] constructed a family of density Θ(1/n 4 ) 1 and Zhao [24] gave a family of density Θ(1/n). The last few years have seen an explosion of papers exploring properties of sum-dominant sets: see [7,10,12,19,20,21,22] for history and overview, [8,14,15,19,24] for explicit constructions, [5,9,13,25] for positive lower bounds for the percentage of sum-dominant sets, [11,16] for generalized sum-dominant sets, and [1,4,6,17,25] for extensions to other settings.…”

Section: Background and Main Resultsmentioning

confidence: 99%

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

Chu

2019

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Given a finite set A ⊆ N, define the sum set A + A = {a i + a j | a i , a j ∈ A} and the difference setThe set A is said to be sum-dominant if |A + A| > |A − A|. We prove the following results 1. The union of two arithmetic progressions (with the same common difference)is not sum-dominant. This result partially proves a conjecture proposed by the author in a previous paper; that is, the union of any two arbitrary arithmetic progressions is not sum-dominant. The author is motivated by the anonymous referee's comment that the conjecture was marvelous and tantalizing.2. Hegarty proved that a sum-dominant set must have at least 8 elements with computers' help. The author of the current paper provided a human-verifiable proof that a sum-dominant set must have at least 7 elements. A natural question is about the largest cardinality of sum-dominant subsets of an arithmetic progression.Fix n ≥ 16. Let N be the cardinality of the largest sum-dominant subset(s) of {0, 1, . . . , n − 1} that contain(s) 0 and n − 1. Then n − 7 ≤ N ≤ n − 4; that is, from an arithmetic progression of length n ≥ 16, we need to discard at least 4 and at most 7 elements (in a clever way) to have the largest sum-dominant set(s).3. Let R ∈ N have the property that for all r ≥ R, {1, 2, . . . , r} can be partitioned into 3 sum-dominant subsets, while {1, 2, . . . , R − 1} cannot. Then 24 ≤ R ≤ 145. This result answers a question by the author et al. in another paper on whether we can find a stricter upper bound for R. The previous upper bound was at least 888.

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Section: Background and Main Resultsmentioning

confidence: 99%

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

Chu

2019

Preprint

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“…Steve Miller and his students and colleagues have contributed greatly to this subject (cf. [2,3,8,9,12,13,14,30,29,31]).…”

Section: Problem 2 a Fundamental Problem Is To Classify The Possible ...mentioning

confidence: 99%

Problems in additive number theory, V: Affinely inequivalent MSTD sets

Nathanson

2016

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“…As addition is commutative and subtraction is not, it was expected that in the limit almost all sets would be difference-dominant, though there were many constructions of infinite families of MSTD sets. 1 There is an extensive literature on such sets, their constructions, and generalizations to settings other than subsets of the integers; see for example [AMMS,BELM,CLMS,CMMXZ,DKMMW,He,HLM,ILMZ,Ma,MOS,MS,MPR,MV,Na1,Na2,PW,Ru1,Ru2,Ru3,Sp,Zh1].…”

Section: Introductionmentioning

confidence: 99%

Generalizing the Distribution of Missing Sums in Sumsets

Chu¹,

King²,

Luntzlara³

et al. 2020

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We examine |A + A| as a random variable, where A ⊂ In = [0, n − 1], the set of integers from 0 to n − 1, so that each element of In is in A with a fixed probability p ∈ (0, 1). Recently, Martin and O'Bryant studied the case in which p = 1/2 and found a closed form for E[|A + A|]. Lazarev, Miller, and O'Bryant extended the result to find a numerical estimate for Var(|A + A|) and bounds on mn ; p(k) := P(2n − 1 − |A + A| = k). Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for E [|A+A|] and Var(|A + A|) for all p ∈ (0, 1) and establish good bounds for E[|A + A|] and mn ; p(k).We continue to investigate mn ; p(k) by studying mp(k) = limn→∞ mn ; p(k), proven to exist by Zhao. Lazarev, Miller, and O'Bryant proved that, for p = 1/2, m 1/2 (6) > m 1/2 (7) < m 1/2 (8). This distribution is not unimodal, and is said to have a "divot" at 7. We report results investigating this divot as p varies, and through both theoretical and numerical analysis prove that for p ≥ 0.68 there is a divot at 1; that is,Finally, we extend the graph-theoretic framework originally introduced by Lazarev, Miller, and O'Bryant to correlated sumsets A + B where B is correlated to A by the probabilitiesWe provide some preliminary results using the extension of this framework. CONTENTS 1. Introduction 2. Generalizations of [MO] 3. Graph-Theoretic Framework 4. Expected Value 5. Variance 6. Divot Computations 6.1. An Upper Bound on m p (k) 6.2. A Lower Bound on m p (k) 7.

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

Cited by 9 publications

References 12 publications

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

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

Problems in additive number theory, V: Affinely inequivalent MSTD sets

Generalizing the Distribution of Missing Sums in Sumsets

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