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A basis of ℤₘ, II

Min Tang, Yong-Gao Chen (2007)

Colloquium Mathematicae

Given a set A ⊂ ℕ let σ A ( n ) denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, σ A ( n ) is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and σ A ( n ̅ ) 5120 for all n̅ ∈ ℤₘ.

A basis of Zₘ

Min Tang, Yong-Gao Chen (2006)

Colloquium Mathematicae

Let σ A ( n ) = | ( a , a ' ) A ² : a + a ' = n | , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, σ A ( n ) is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which σ A ( n ) is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and σ A ( n ̅ ) 768 for all n̅ ∈ Zₘ.

A generalization of a theorem of Erdös on asymptotic basis of order 2

Martin Helm (1994)

Journal de théorie des nombres de Bordeaux

Let 𝒯 be a system of disjoint subsets of * . In this paper we examine the existence of an increasing sequence of natural numbers, A , that is an asymptotic basis of all infinite elements T j of 𝒯 simultaneously, satisfying certain conditions on the rate of growth of the number of representations 𝑟 𝑛 ( 𝐴 ) ; 𝑟 𝑛 ( 𝐴 ) : = ( 𝑎 𝑖 , 𝑎 𝑗 ) : 𝑎 𝑖 < 𝑎 𝑗 ; 𝑎 𝑖 , 𝑎 𝑗 𝐴 ; 𝑛 = 𝑎 𝑖 + 𝑎 𝑗 , for all sufficiently large n T j and j * A theorem of P. Erdös is generalized.

A note on sumsets of subgroups in * p

Derrick Hart (2013)

Acta Arithmetica

Let A be a multiplicative subgroup of * p . Define the k-fold sumset of A to be k A = x 1 + . . . + x k : x i A , 1 i k . We show that 6 A * p for | A | > p 11 / 23 + ϵ . In addition, we extend a result of Shkredov to show that | 2 A | | A | 8 / 5 - ϵ for | A | p 5 / 9 .

A propos de la fonction X d’Erdös et Graham

Alain Plagne (2004)

Annales de l’institut Fourier

Nous améliorons les meilleures bornes supérieures et inférieures connues pour la fonction X d’Erdös et Graham définie par X ( h ) = max h 𝒜 max a 𝒜 * ord * ( 𝒜 a ) , où le premier maximum est pris sur toutes les bases (exactes) 𝒜 d’ordre au plus h , où 𝒜 * désigne le sous-ensemble de 𝒜 composé des éléments a tels que 𝒜 { a } soit encore une base et où, enfin, ord * ( 𝒜 ) désigne l’ordre (exact) de 𝒜 . Notre étude nous conduira, entre autres, à prouver un nouveau résultat additif général découlant de la méthode isopérimétrique et à étudier trois problèmes additifs...

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