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Given a set A ⊂ ℕ let 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, 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 for all n̅ ∈ ℤₘ.
Let , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which 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 for all n̅ ∈ Zₘ.
Let be a system of disjoint subsets of . In this paper we examine the existence of an increasing sequence of natural numbers, , that is an asymptotic basis of all infinite elements of simultaneously, satisfying certain conditions on the rate of growth of the number of representations , for all sufficiently large and A theorem of P. Erdös is generalized.
Let A be a multiplicative subgroup of . Define the k-fold sumset of A to be . We show that for . In addition, we extend a result of Shkredov to show that for .
Nous améliorons les meilleures bornes supérieures et inférieures connues pour la fonction
d’Erdös et Graham définie par , où le premier maximum est pris sur
toutes les bases (exactes) d’ordre au plus , où désigne le
sous-ensemble de composé des éléments tels que soit encore une base et où, enfin, désigne l’ordre (exact) de
. 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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