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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
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References:

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