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**Zentralblatt MATH**

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

**Zbl.No: ** 608.10050

**Autor: ** Erdös, Paul; Nathanson, Melvyn B.

**Title: ** Independence of solution sets in additive number theory. (In English)

**Source: ** Adv. Math., Suppl. Stud. 9, 97-105 (1986).

**Review: ** Let A\subseteq **N**, then A is called an asymptotic basis of order 2 if for all sufficiently large n in **N** there are a,a' in A such that n = a+a'. Let S_{A}(n) = **{**a in A| n-a in A, n\ne 2a**}** denote the solution set of n. By the Erdös-Rényi probabilistic method [see *H.Halberstam* and *K.F.Roth*, Sequences (1966; Zbl 141.04405), p. 141 ff.] it is shown that for almost all A in the space \Omega of all strictly increasing sequences of positive integers the cardinality of S_{A}(m)\cap S_{A}(n) is bounded for all m < n. The bound depends on the chosen probability measure on \Omega only. This result is useful to proof the existence of minimal asymptotic bases A of order 2, which means A has no proper subset being an asymptotic basis of order 2 itself. It is proved that A\subseteq **N** contains a minimal asymptotic basis of order 2 if |S_{A}(m)\cap S_{A}(n)| is bounded for all m < n and **lim**_{n ––> oo}|S_{A}(n)| = oo.

**Reviewer: ** J.Zöllner

**Classif.: ** * 11B13 Additive bases

11K99 Probabilistic theory

**Keywords: ** asymptotic basis; solution set; Erdös-Rényi probabilistic method; existence of minimal asymptotic bases

**Citations: ** Zbl 141.044

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