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**Zentralblatt MATH**

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

**Zbl.No: ** 426.10057

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

**Title: ** Minimal asymptotic bases for the natural numbers. (In English)

**Source: ** J. Number Theory 12, 154-159 (1980).

**Review: ** The sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A. Let M_{h}^{A} denote the et of elements that have more than one representation as a sum of h elements of A. It is proved that there exists an asymptotic basis A such that M_{h}^{A}(x) = 0(x^{1-1/h+\epsilon}) for every \epsilon > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2,3, and 4. Several open problems concerning minimal bases are also discussed.

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

**Keywords: ** minimal asymptotic bases

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