no. 1
A generalization of a theorem of Erdös on asymptotic basis of order 2
Journal de Théorie des Nombres de Bordeaux, Tome 6 (1994) no. 1, pp. 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 (𝐴);𝑟 𝑛 (𝐴):=(𝑎 𝑖 ,𝑎 𝑗 ):𝑎 𝑖 <𝑎 𝑗 ;𝑎 𝑖 ,𝑎 𝑗 ∈𝐴;𝑛=𝑎 𝑖 +𝑎 𝑗 , for all sufficiently large n∈T j and j∈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},
     pages = {9--19},
     publisher = {Universit\'e Bordeaux I},
     volume = {6},
     number = {1},
     year = {1994},
     zbl = {0812.11011},
     mrnumber = {1305285},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1994__6_1_9_0/}
}
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
DA  - 1994///
SP  - 9
EP  - 19
VL  - 6
IS  - 1
PB  - Université Bordeaux I
UR  - http://www.numdam.org/item/JTNB_1994__6_1_9_0/
UR  - https://zbmath.org/?q=an%3A0812.11011
UR  - https://www.ams.org/mathscinet-getitem?mr=1305285
LA  - en
ID  - JTNB_1994__6_1_9_0
ER  - 
Helm, Martin. A generalization of a theorem of Erdös on asymptotic basis of order $2$. Journal de Théorie des Nombres de Bordeaux, Tome 6 (1994) no. 1, pp. 9-19. http://www.numdam.org/item/JTNB_1994__6_1_9_0/

[1] P. Erdös, Problems and results in additive number theory, Colloque sur la Théorie des Nombres (CBRM), Bruxelles (1956), 127-137. | MR 79027 | Zbl 0073.03102

[2] P. Erdös and A. Rényi, Additive properties of random sequences of positive integers, Acta Arith. 6 (1960), 83-110. | MR 120213 | Zbl 0091.04401

[3] H. Halberstam and K.F. Roth, Sequences, Springer-Verlag, New-York Heidelberg Berlin (1983). | MR 687978 | Zbl 0498.10001

[4] I.Z. Rusza, On a probabilistic method in additive number theory, Groupe de travail en théorie analytique et élémentaire des nombres, (1987-1988), Publications Mathématiques d'Orsay 89-01, Univ. Paris, Orsay (1989), 71-92. | MR 993303 | Zbl 0672.10037

[5] S. Sidon, Ein Satz über trigonometrische Polynorne und seine Anwendung in der Theorie des Fourier-Reihen, Math. Ann. 106 (1932), 539-539. | JFM 58.0268.06 | Zbl 0004.21203