A Pollard Type Result for Restricted Sums

Caldeira, Cristina; Silva, J.A Dias da

doi:10.1006/jnth.1998.2269

Cited by 6 publications

(4 citation statements)

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“…Caldeira and Dias da Silva gave in [2] an extension of the Pollard's theorem to an arbitrary field, and in [1] an analogue for restricted sums.…”

Section: B) T \ > Min(«p T(\a\ + \B\ -T))mentioning

confidence: 99%

Blocks of consecutive integers in sumsets (A + B)_t

Guo¹,

Chen²

2004

Bull. Austral. Math. Soc.

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Let A,B C {1,..., n}. For m G Z, let r AtB {m) be the cardinality of the set of ordered pairs (a, b) 6 A x B such that a + b = m. For t ^ 1, denote by (A + B) t the set of the elements m for which r^si'm) ^ t. In this paper we prove that for any subsets A,B C {l,...,n} such that |v4| + | 5 | ^ (4n + 4 t -3 ) / 3 , the sumset (A + B) t contains a block of consecutive integers with the length at least \A\ + \B\ -2t +1, and that (a) for any two subsets A and B of {1,..., n} such that \A\ + \B\ ^ (4n)/3, there exists an arithmetic progression of length n in A + B; (b) for any 2 $C r < (An -l)/3, there exist two subsets A and B of {1,... , n} with \A\ + \B\ = r such that any arithmetic progression in A + B has the length at most (2n -l)/3 + 1.

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“…Caldeira and Dias da Silva gave in [2] an extension of the Pollard's theorem to an arbitrary field, and in [1] an analogue for restricted sums.…”

Section: B) T \ > Min(«p T(\a\ + \B\ -T))mentioning

confidence: 99%

Blocks of consecutive integers in sumsets (A + B)_t

Guo¹,

Chen²

2004

Bull. Austral. Math. Soc.

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“…Very recently Nazarewicz, O'Brien, O'Neill and C. Staples [12] obtained the equality cases in Pollard's Theorem A. A restricted-sum version of the inequalities is obtained by Caldeira and Dias da Silva [1].…”

Section: Introductionmentioning

confidence: 98%

A note on Pollard's Theorem

Hamidoune,

Serra

2008

Preprint

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Let A, B be nonempty subsets of a an abelian group G. Let N i (A, B) denote the set of elements of G having i distinct decompositions as a product of an element of A and an element of B. We prove thatwhere α is the largest size of a coset contained in AB and w = min(α − 1, 1), with a strict inequality if α ≥ 3 and t ≥ 2, or if α ≥ 2 and t = 2.This result is a local extension of results by Pollard and Green-Ruzsa and extends also for t > 2 a recent result of Grynkiewicz, conjectured by Dicks-Ivanov (for non necessarily abelian groups) in connection to the famous Hanna Neumann problem in Group Theory.

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“…An extension for restricted sumsets over fields is also known [1] [5]. However, very little is known concerning t-representable sums in an arbitrary abelian group.…”

Section: Introductionmentioning

confidence: 99%

“…His bound is tight as is seen by considering two arithmetic progressions with the same difference, and in fact the cases of equality have been characterized [17]. An extension for restricted sumsets over fields is also known [1] [5]. However, very little is known concerning t-representable sums in an arbitrary abelian group.…”

Section: Introductionmentioning

confidence: 99%

On extending Pollard’s theorem for t-representable sums

Grynkiewicz

2010

Isr. J. Math.

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Abstract. Let t ≥ 1, let A and B be finite, nonempty subsets of an abelian group G, and let A + i B denote all the elements c with at least i representations of the form c = a + b, with a ∈ A and b ∈ B. For |A|, |B| ≥ t, we show that eitheror else there exist A ′ ⊆ A and B ′ ⊆ B withwhere H is the (nontrivial) stabilizer of A + t B and ρ = |A ′ + H| − |A ′ | + |B ′ + H| − |B ′ |.In the case t = 2, we improve (1) to |A + 1 B| + |A + 2 B| ≥ 2|A| + 2|B| − 4.

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A Pollard Type Result for Restricted Sums

Cited by 6 publications

References 5 publications

Blocks of consecutive integers in sumsets (A + B)_t

Blocks of consecutive integers in sumsets (A + B)_t

A note on Pollard's Theorem

On extending Pollard’s theorem for t-representable sums

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A Pollard Type Result for Restricted Sums

Cited by 6 publications

References 5 publications

Blocks of consecutive integers in sumsets (A + B)t

Blocks of consecutive integers in sumsets (A + B)t

A note on Pollard's Theorem

On extending Pollard’s theorem for t-representable sums

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Product

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Blocks of consecutive integers in sumsets (A + B)_t

Blocks of consecutive integers in sumsets (A + B)_t