A just basis |
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Authors: | Imre Z. Ruzsa |
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Affiliation: | 1. Mathematical Institute of the Hungarian Academy of Sciences, Pf. 127, H-1364, Budapest, Hungary
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Abstract: | An old problem of P. Erdös and P. Turán asks whether there is a basisA of order 2 for which the number of representationsn=a+a′, a,a′∈A is bounded. Erd?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, a t ) ∈A 2:a+a′=n}| we prove that there exists a setA of nonnegative integers that forms a basis of order 2 (that is,s(n)≥1 for alln), and satisfies ∑n ? N σ(N)2 = O(N). |
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