The Haight–Ruzsa Method for Sets with More Differences than Multiple Sums

Nathanson, Melvyn B.

doi:10.1017/9781316650295.011

Cited by 2 publications

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“…, t h ) = 2t 1 + h i=2 t i also satisfy the conditions of Theorem 2. This answers a question in [3].…”

Section: Resultsmentioning

confidence: 85%

Comparison estimates for linear forms in additive number theory

Nathanson

2018

Journal of Number Theory

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Abstract. Let R be a commutative ring R with 1 R and with group of units R × . Let Φ = Φ(t 1 , . . . , t h ) = h i=1 ϕ i t i be an h-ary linear form with nonzero coefficients ϕ 1 , . . . , ϕ h ∈ R. Let M be an R-module. For every subset A of M , the image of A under Φ isFor every subset I of {1, 2, . . . , h}, there is the subset sum s I = i∈I ϕ i . Let S(Φ) = {s I : ∅ = I ⊆ {1, 2, . . . , h}}. The difference set A − A is the image of A under the linear form Υ(t 1 , t 2 ) = t 1 − t 2 and the h-fold sumset hA is the image of A under the linear form Φ(t 1 , t 2 , . . . , t h ) = t 1 + t 2 + · · · + t h . Equivalently, Ruzsa constructed a subset A of the Z-module M = Z/mZ such thatThis is a significant result in additive number theory. In this paper, we extend Ruzsa's theorem to a large class of pairs of linear forms Υ and Φ. Let R be a commutative ring with multiplicative identity 1 R = 0. We denote the group of units in R by R × . Associated to every sequence (ϕ 1 , . . . , ϕ h ) of nonzero

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“…, t h ) = 2t 1 + h i=2 t i also satisfy the conditions of Theorem 2. This answers a question in [3].…”

Section: Resultsmentioning

confidence: 85%

Comparison estimates for linear forms in additive number theory

Nathanson

2018

Journal of Number Theory

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“…Let ǫ > 0 be any small real. Then, we can find q, a product of arbitrarily large distinct primes and maps α, β, γ: Zq → Zq such that the expression (9) takes at most ǫq values in Zq.…”

Section: Discussionmentioning

confidence: 99%

“…Finally, we construct the desired maps for this expression. By adding linear terms to α, β, γ, we may assume that the expression is α(x)β(y) + (α(x) + c1x)(β(y) + c2y) + α(x)γ(z) + λ1α(x) + µ1x + λ2β(y) + µ2y + λ3γ(z) + µ3z (9) for some coefficients c1, c2 ∈ {−1, 1}, λ1, λ2, λ3 ∈ N0, µ1, µ2, µ3 ∈ Z. Let us begin by observing that in most cases there is a rather simple solution, which strangely we could not generalize to work for all choices of coefficients.…”

Section: E Has Three Variablesmentioning

confidence: 99%

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Small Sets with Large Difference Sets

Milićević¹

2017

Preprint

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For every ǫ > 0 and k ∈ N, Haight constructed a set A ⊂ ZN (ZN stands for the integers modulo N ) for a suitable N , such that A − A = ZN and |kA|< ǫN . Recently, Nathanson posed the problem of constructing sets A ⊂ ZN for given polynomials p and q, such that p(A) = ZN and |q(A)|< ǫN , where p(A) is the set {p(a1, a2, . . . , an): a1, a2, . . . , an ∈ A}, when p has n variables. In this paper, we give a partial answer to Nathanson's question. For every k ∈ N and ǫ > 0, we find a set A ⊂ ZN for suitable N , such that A − A = ZN , but |A 2 + kA|< ǫN , whereWe also extend this result to construct, for every k ∈ N and ǫ > 0, a set A ⊂ ZN for suitable N , such that A − A = ZN , but |3A 2 + kA|< ǫN , where 3A 2 + kA = {a1a2 + a3a4 + a5a6 + b1 + b2 + • • • + b k : a1, . . . , a6, b1, . . . , b k ∈ A}.

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The Haight–Ruzsa Method for Sets with More Differences than Multiple Sums

Cited by 2 publications

References 5 publications

Comparison estimates for linear forms in additive number theory

Comparison estimates for linear forms in additive number theory

Small Sets with Large Difference Sets

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