## A generalization of a theorem of Erdős on asymptotic basis of order 2.(English)Zbl 0812.11011

A classical theorem of Erdős asserts the existence of a basis of order 2 such that the number of representations satisfies $$r(n) \asymp \log n$$. Here the following analogue is proved. For every $$k$$ there is a set $$A$$ and a decomposition of the set of integers into disjoint sets $$T_ j$$ such that the counting function of each $$T_ j$$ is of order $$\log_{k- 1} n$$, while $$r(n)$$, the number of representations of $$n$$ as a sum of two elements of $$A$$, satisfies $$r(n) \asymp \log_ k n$$ on each $$T_ j$$, that is, for each $$j$$ the set $\{n\in T_ j:\;r(n)\not\in (c_ 1\log_ k n, c_ 2\log_ k n)\}$ is finite. ($$\log_ k n$$ is the $$k$$ times iterated logarithm).

### MSC:

 11B13 Additive bases, including sumsets 11B83 Special sequences and polynomials

### Keywords:

asymptotic basis of order 2; additive bases
Full Text:

### References:

 [1] Erdös, P., Problems and results in additive number theory, Colloque sur la Théorie des Nombres (CBRM), Bruxelles (1956), 127-137. · Zbl 0073.03102 [2] Erdös, P. and Rényi, A., Additive properties of random sequences of positive integers, Acta Arith.6 (1960), 83-110. · Zbl 0091.04401 [3] Halberstam, H. and Roth, K.F., Sequences, Springer-Verlag, New-YorkHeidelbergBerlin (1983). · Zbl 0498.10001 [4] Rusza, I.Z., 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. · Zbl 0672.10037 [5] Sidon, S., Ein Satz über trigonometrische Polynorne und seine Anwendung in der Theorie des Fourier-Reihen, Math. Ann.106 (1932), 539-539. · JFM 58.0268.06
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.