### 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 |

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### Keywords

- Sidon sequences
- The probabilistic method

### ASJC Scopus subject areas

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

### Cite this

*Random Structures and Algorithms*,

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

**Generalization of a theorem of Erdo{double acute}s and Rényi on Sidon sequences.** / Cilleruelo, Javier; Kiss, Sándor Z.; Ruzsa, I.; Vinuesa, Carlos.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 37, no. 4, pp. 455-464. https://doi.org/10.1002/rsa.20350

}

TY - JOUR

T1 - Generalization of a theorem of Erdo{double acute}s and Rényi on Sidon sequences

AU - Cilleruelo, Javier

AU - Kiss, Sándor Z.

AU - Ruzsa, I.

AU - Vinuesa, Carlos

PY - 2010/12/8

Y1 - 2010/12/8

N2 - Erdo{double acute}s and Rényi claimed and Vu proved that for all h ≥ 2 and for all h > 0, there exists g = gh(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]| ≫ x1/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 gh(ε) ≪ ε-1, improving the bound gh(ε) ≪ ε-h+1 obtained by Vu.Finally we use the "alteration method" to get a better bound for g3(ε), obtaining a more precise estimate for the growth of B3[g] sequences.

AB - Erdo{double acute}s and Rényi claimed and Vu proved that for all h ≥ 2 and for all h > 0, there exists g = gh(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]| ≫ x1/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 gh(ε) ≪ ε-1, improving the bound gh(ε) ≪ ε-h+1 obtained by Vu.Finally we use the "alteration method" to get a better bound for g3(ε), obtaining a more precise estimate for the growth of B3[g] sequences.

KW - Sidon sequences

KW - The probabilistic method

UR - http://www.scopus.com/inward/record.url?scp=77958533657&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77958533657&partnerID=8YFLogxK

U2 - 10.1002/rsa.20350

DO - 10.1002/rsa.20350

M3 - Article

AN - SCOPUS:77958533657

VL - 37

SP - 455

EP - 464

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 4

ER -