A just basis

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An old problem of P. Erdös and P. Turán asks whether there is a basis A of order 2 for which the number of representations n=a+a′, a,a′∈A is bounded. Erdo{combining double acute accent}s conjectured that such a basis does not exist. We answer a related finite problem and find a basis for which the number of representations is bounded in the square mean. Writing σ (n)=|{(a, at) ∈A2:a+a′=n}| we prove that there exists a set A of nonnegative integers that forms a basis of order 2 (that is, s(n)≥1 for all n), and satisfies ∑n ≤ N σ(N)2 = O(N).

Original languageEnglish
Pages (from-to)145-151
Number of pages7
JournalMonatshefte für Mathematik
Issue number2
Publication statusPublished - Jun 1 1990

ASJC Scopus subject areas

  • Mathematics(all)

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