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Minimal asymptotic bases for the natural numbers

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Publication Date
Volume
12
Issue
2
Identifiers
DOI: 10.1016/0022-314x(80)90048-7

Abstract

Abstract 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 set 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) = O(x 1−1 h+ϵ ) for every ϵ > 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.

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