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

Helm, Martin. "A generalization of a theorem of Erdös on asymptotic basis of order $2$." Journal de théorie des nombres de Bordeaux 6.1 (1994): 9-19. <http://eudml.org/doc/247530>.

@article{Helm1994,
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 $\it \{r\} _n ( A); \it \{r\} _n(A) :=\left|\left\lbrace (a_i,a_j): a_i &lt; a_j; a_i, a_j \in A; n = a_i + a_j \right\rbrace \right|$, for all sufficiently large $n \in T_j$ and $j \in \mathbb \{N\}^*$ A theorem of P. Erdös is generalized.},
author = {Helm, Martin},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {asymptotic basis of order 2; additive bases},
language = {eng},
number = {1},
pages = {9-19},
publisher = {Université Bordeaux I},
title = {A generalization of a theorem of Erdös on asymptotic basis of order $2$},
url = {http://eudml.org/doc/247530},
volume = {6},
year = {1994},
}

TY - JOUR
AU - Helm, Martin
TI - A generalization of a theorem of Erdös on asymptotic basis of order $2$
JO - Journal de théorie des nombres de Bordeaux
PY - 1994
PB - Université Bordeaux I
VL - 6
IS - 1
SP - 9
EP - 19
AB - 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 $\it {r} _n ( A); \it {r} _n(A) :=\left|\left\lbrace (a_i,a_j): a_i &lt; a_j; a_i, a_j \in A; n = a_i + a_j \right\rbrace \right|$, for all sufficiently large $n \in T_j$ and $j \in \mathbb {N}^*$ A theorem of P. Erdös is generalized.
LA - eng
KW - asymptotic basis of order 2; additive bases
UR - http://eudml.org/doc/247530
ER -