## 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 MhA 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

MhA(x) = 0(x1-1/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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