### Abstract

Erdo{double acute}s and Rényi claimed and Vu proved that for all h ≥ 2 and for all h > 0, there exists g = g_{h}(ε) and a sequence of integers A such that the number of ordered representations of any number as a sum of h elements of A is bounded by g, and such that |A ∩ [1,x]| ≫ x^{1/h-ε}.We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a g that satisfies g_{h}(ε) ≪ ε^{-1}, improving the bound g_{h}(ε) ≪ ε^{-h+1} obtained by Vu.Finally we use the "alteration method" to get a better bound for g_{3}(ε), obtaining a more precise estimate for the growth of B_{3}[g] sequences.

Original language | English |
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Pages (from-to) | 455-464 |

Number of pages | 10 |

Journal | Random Structures and Algorithms |

Volume | 37 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 8 2010 |

### Keywords

- Sidon sequences
- The probabilistic method

### ASJC Scopus subject areas

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

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## Cite this

*Random Structures and Algorithms*,

*37*(4), 455-464. https://doi.org/10.1002/rsa.20350