A generalization of a theorem of Erdös on asymptotic basis of order $2$
Journal de théorie des nombres de Bordeaux, Volume 6 (1994) no. 1, p. 9-19

Let $𝒯$ be a system of disjoint subsets of ${ℕ}^{*}$. 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 ${𝑟}_{𝑛}\left(𝐴\right);{𝑟}_{𝑛}\left(𝐴\right):=\left|\left\{\left({𝑎}_{𝑖},{𝑎}_{𝑗}\right):{𝑎}_{𝑖}<{𝑎}_{𝑗};{𝑎}_{𝑖},{𝑎}_{𝑗}\in 𝐴;𝑛={𝑎}_{𝑖}+{𝑎}_{𝑗}\right\}\right|$, for all sufficiently large $n\in {T}_{j}$ and $j\in {ℕ}^{*}$ A theorem of P. Erdös is generalized.

@article{JTNB_1994__6_1_9_0,
author = {Helm, Martin},
title = {A generalization of a theorem of Erd\"os on asymptotic basis of order $2$},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux I},
volume = {6},
number = {1},
year = {1994},
pages = {9-19},
zbl = {0812.11011},
mrnumber = {1305285},
language = {en},
url = {http://www.numdam.org/item/JTNB_1994__6_1_9_0}
}

Helm, Martin. A generalization of a theorem of Erdös on asymptotic basis of order $2$. Journal de théorie des nombres de Bordeaux, Volume 6 (1994) no. 1, pp. 9-19. http://www.numdam.org/item/JTNB_1994__6_1_9_0/

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