### Abstract

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, a^{t}) ∈A^{2}: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 language | English |
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Pages (from-to) | 145-151 |

Number of pages | 7 |

Journal | Monatshefte für Mathematik |

Volume | 109 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 1990 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Ruzsa, I. Z. (1990). A just basis.

*Monatshefte für Mathematik*,*109*(2), 145-151. https://doi.org/10.1007/BF01302934