### Abstract

If S is an arbitrary sequence of positive integers, let P(S) be the set of all integers which are representable as a sum of distinct terms of S. Call S complete if P(S) contains all large integers, and subcomplete if P(S) contains an infinite arithmetic progression. It is shown that any sequence can be perturbed in a rather moderate way into a sequence which is not subcomplete. On the other hand, it is shown that if S is any sequence satisfying a mild growth condition, then a surprisingly gentle perturbation suffices to make S complete in a strong sense. Various related questions are also considered.

Original language | English |
---|---|

Pages (from-to) | 446-455 |

Number of pages | 10 |

Journal | Journal of Number Theory |

Volume | 13 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1981 |

### Fingerprint

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*13*(4), 446-455. https://doi.org/10.1016/0022-314X(81)90036-6

**Completeness properties of perturbed sequences.** / Burr, Stefan A.; Erdős, P.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 13, no. 4, pp. 446-455. https://doi.org/10.1016/0022-314X(81)90036-6

}

TY - JOUR

T1 - Completeness properties of perturbed sequences

AU - Burr, Stefan A.

AU - Erdős, P.

PY - 1981

Y1 - 1981

N2 - If S is an arbitrary sequence of positive integers, let P(S) be the set of all integers which are representable as a sum of distinct terms of S. Call S complete if P(S) contains all large integers, and subcomplete if P(S) contains an infinite arithmetic progression. It is shown that any sequence can be perturbed in a rather moderate way into a sequence which is not subcomplete. On the other hand, it is shown that if S is any sequence satisfying a mild growth condition, then a surprisingly gentle perturbation suffices to make S complete in a strong sense. Various related questions are also considered.

AB - If S is an arbitrary sequence of positive integers, let P(S) be the set of all integers which are representable as a sum of distinct terms of S. Call S complete if P(S) contains all large integers, and subcomplete if P(S) contains an infinite arithmetic progression. It is shown that any sequence can be perturbed in a rather moderate way into a sequence which is not subcomplete. On the other hand, it is shown that if S is any sequence satisfying a mild growth condition, then a surprisingly gentle perturbation suffices to make S complete in a strong sense. Various related questions are also considered.

UR - http://www.scopus.com/inward/record.url?scp=17644385120&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17644385120&partnerID=8YFLogxK

U2 - 10.1016/0022-314X(81)90036-6

DO - 10.1016/0022-314X(81)90036-6

M3 - Article

VL - 13

SP - 446

EP - 455

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 4

ER -