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**Zentralblatt MATH**

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

**Zbl.No: ** 669.10078

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

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

**Source: ** Stud. Sci. Math. Hung. 22, No.1-4, 53-63 (1987).

**Review: ** Let A = **{**a_{1} < a_{2} < ...**}** be an infinite sequence of positive integers, A(N) be its counting function and R(n) be the number of representations of n as a sum of two elements of A. In parts I and II [cf. Pac. J. Math. 118, 347-357 (1985; Zbl 569.10032), Acta Math. Hung. 48, 201-211 (1986; Zbl 621.10041)] the first two authors investigated how well R(n) is approximable by ``nice'' functions. In this paper the boundedness of |R(n+1)-R(n)| is studied. A simple example shows that the density A(N) of A is not relevant in this question. Rather the number of blocks determines the answer. Let B(N) be the number of blocks with starting elements up to N. It turns out that if B(N)/N^{ ½} tends to infinity then |R(n+1)-R(n)| can not be bounded, while there are examples of sets A with B(N) >> N^{-\epsilon} and R(n) itself is bounded.

[Parts IV and V were published (with *V. Turán Sós*) in Lect. Notes Math. 1122, 85-104 (1985; Zbl 588.10056) and Monatsh. Math. 102, 183-197 (1986; Zbl 597.10055).]

**Reviewer: ** A.Balog

**Classif.: ** * 11B83 Special sequences of integers and polynomials

11B34 Representation functions

11B13 Additive bases

00A07 Problem books

**Keywords: ** sum of sequences; infinite sequence of positive integers; counting function; number of representations; sum of two elements; number of blocks

**Citations: ** Zbl 569.10032; Zbl 621.10041; Zbl 588.10056; Zbl 597.10055

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