### Abstract

We employ the probabilistic method to prove a stronger version of a result of Helm, related to a conjecture of Erdo{combining double acute accent}s and Turán about additive bases of the positive integers. We show that for a class of random sequences of positive integers A, which satisfy ∣A ∩ [1, x]∣ ≫ √x with probability 1, all integers in any interval [ I, N] can be written in at least c_{1} log N and at most c_{2} log N ways as a difference of elements of A ∩ [1, N^{2}]. We also prove several results related to another result of Helm. We show that for every sequence of positive integers M, with counting function M(x), there is always another sequence of positive integers A such that M ∩ (A - A) = ∅ and A(x) > x/(M(x) + 1). We also show that this result is essentially best possible, and we show how to construct a sequence A with A(x) > cx/(M(x) + 1) for which every element of M is represented exactly as many times as we wish as a difference of elements of A.

Original language | English (US) |
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Pages (from-to) | 82-93 |

Number of pages | 12 |

Journal | Journal of Number Theory |

Volume | 55 |

Issue number | 1 |

DOIs | |

State | Published - Nov 1995 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

*Journal of Number Theory*,

*55*(1), 82-93. https://doi.org/10.1006/jnth.1995.1129