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 SA(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 SA(m)\cap SA(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 |SA(m)\cap SA(n)| is bounded for all m < n and limn ––> oo|SA(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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