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

Javier Cilleruelo, Sándor Z. Kiss, I. Ruzsa, Carlos Vinuesa

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)455-464
Number of pages10
JournalRandom Structures and Algorithms
Volume37
Issue number4
DOIs
Publication statusPublished - Dec 8 2010

Fingerprint

Acute
Theorem
Integer
Estimate
Generalization

Keywords

  • Sidon sequences
  • The probabilistic method

ASJC Scopus subject areas

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

Cite this

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

In: Random Structures and Algorithms, Vol. 37, No. 4, 08.12.2010, p. 455-464.

Research output: Contribution to journalArticle

Cilleruelo, Javier ; Kiss, Sándor Z. ; Ruzsa, I. ; Vinuesa, Carlos. / Generalization of a theorem of Erdo{double acute}s and Rényi on Sidon sequences. In: Random Structures and Algorithms. 2010 ; Vol. 37, No. 4. pp. 455-464.
@article{bc149f68a8364671bf5174f7e44bd2a9,
title = "Generalization of a theorem of Erdo{double acute}s and R{\'e}nyi on Sidon sequences",
abstract = "Erdo{double acute}s and R{\'e}nyi 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.",
keywords = "Sidon sequences, The probabilistic method",
author = "Javier Cilleruelo and Kiss, {S{\'a}ndor Z.} and I. Ruzsa and Carlos Vinuesa",
year = "2010",
month = "12",
day = "8",
doi = "10.1002/rsa.20350",
language = "English",
volume = "37",
pages = "455--464",
journal = "Random Structures and Algorithms",
issn = "1042-9832",
publisher = "John Wiley and Sons Ltd",
number = "4",

}

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 -