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

Helm, Martin

Journal de théorie des nombres de Bordeaux, Tome 6 (1994) no. 1, pp. 9-19

Résumé

Let $𝒯$ 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 $𝒯$ simultaneously, satisfying certain conditions on the rate of growth of the number of representations ${𝑟}_{𝑛}\left(𝐴\right);{𝑟}_{𝑛}\left(𝐴\right):=\left|\left\{({𝑎}_{𝑖},{𝑎}_{𝑗}):{𝑎}_{𝑖}<{𝑎}_{𝑗};{𝑎}_{𝑖},{𝑎}_{𝑗}\in 𝐴;𝑛={𝑎}_{𝑖}+{𝑎}_{𝑗}\right\}\right|$, for all sufficiently large $n\in T_j$ and $j\in {\mathbb{N}}^{*}$ A theorem of P. Erdös is generalized.