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Martin Helm
A generalization of a theorem of Erdös on asymptotic basis of order $2$
Journal de théorie des nombres de Bordeaux, 6 no. 1 (1994), p. 9-19, doi: 10.5802/jtnb.103
Article PDF | Analyses MR 1305285 | Zbl 0812.11011

Résumé - Abstract

Let $\mathcal{T}$ be a system of disjoint subsets of $\mathbb{N}^*$. 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 $\mathcal{T}$ simultaneously, satisfying certain conditions on the rate of growth of the number of representations $\it {r} _n ( A); \it {r} _n(A) :=\left|\left\rbrace (a_i,a_j): a_i < a_j; a_i, a_j \in A; n = a_i + a_j \right\lbrace \right|$, for all sufficiently large $n \in T_j$ and $j \in \mathbb{N}^*$ A theorem of P. Erdös is generalized.


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