Minimal asymptotic bases for the natural numbers

Paul Erdös, Melvyn B. Nathanson

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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 Mh 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 Mh A(x) = O(x 1-1 h+ε{lunate}) for every ε{lunate} > 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.

Original languageEnglish
Pages (from-to)154-159
Number of pages6
JournalJournal of Number Theory
Volume12
Issue number2
DOIs
Publication statusPublished - May 1980

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ASJC Scopus subject areas

  • Algebra and Number Theory

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