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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 588.10056

**Autor: ** Erdös, Paul; Sárközy, A.; Sós, V.T.

**Title: ** Problems and results on additive properties of general sequences. IV. (In English)

**Source: ** Number theory, Proc. 4th Matsci. Conf., Ootacamund/India 1984, Lect. Notes Math. 1122, 85-104 (1985).

**Review: ** [For the entire collection see Zbl 547.00014. - Part I, see the first and second author, Pac. J. Math. 118, 347-357 (1985; Zbl 569.10032).]

Let *A* = **{**a_{1} < a_{2} < ...**}** be an infinite sequence of positive integers and R_{1}(n), R_{2}(n), R_{3}(n) denote the number of solutions of a_{x}+a_{y} = n, a_{x} in *A*, a_{y} in *A* in the cases: no restriction, x < y, x \leq y, respectively. It turns out that these functions behave quite different according to monotonicity.

The authors show that R_{1}(n) is monotonous increasing iff *A* consists of all the integers from a point onwards. Denoting the number of elements of *A* up to n by A(n) they construct sequences *A* such that R_{2}(n) is monotonous increasing and A(n) < n- cn^{1/3}. There is no corresponding result for R_{3}(n), however it is proved that R_{3}(n) and R_{2}(n) cannot be monotonous increasing when A(n) = o(n/ log n). The authors conjecture that this is true with A(n) = o(n).

**Reviewer: ** A.Balog

**Classif.: ** * 11P99 Additive number theory

11B13 Additive bases

05B10 Difference sets

00A07 Problem books

**Keywords: ** number of additive representations; infinite sequence of positive integers; monotonicity

**Citations: ** Zbl 547.00014; Zbl 569.10032

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