### Abstract

Let A be a set of n real numbers such that the number of distinct twofold sums is αn. We show that the number of twofold products is ≧ cn^{2}/(α^{4} log n), and the number of quotients is ≧ cn^{2}/min (α^{6}, α ^{4} log n) with some absolute constant c. For bounded α this gives the correct order of magnitude for the quotients. For sums we think that the correct order is n^{2}/(log n)^{a} with some a <1, perhaps with 2 log 2 - 1, as a result of Pomerance and Sárközy suggests. We also give more general inequalities for sums, products and quotients formed with different sets. The proofs use geometric tools, mainly the Szemerédi-Trotter inequality.

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

Pages (from-to) | 301-308 |

Number of pages | 8 |

Journal | Studia Scientiarum Mathematicarum Hungarica |

Volume | 40 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2003 |

### Fingerprint

### Keywords

- Product set
- Sumset

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Studia Scientiarum Mathematicarum Hungarica*,

*40*(3), 301-308. https://doi.org/10.1556/SScMath.40.2003.3.4

**Few sums, many products.** / Elekes, Gy; Ruzsa, I.

Research output: Contribution to journal › Article

*Studia Scientiarum Mathematicarum Hungarica*, vol. 40, no. 3, pp. 301-308. https://doi.org/10.1556/SScMath.40.2003.3.4

}

TY - JOUR

T1 - Few sums, many products

AU - Elekes, Gy

AU - Ruzsa, I.

PY - 2003

Y1 - 2003

N2 - Let A be a set of n real numbers such that the number of distinct twofold sums is αn. We show that the number of twofold products is ≧ cn2/(α4 log n), and the number of quotients is ≧ cn2/min (α6, α 4 log n) with some absolute constant c. For bounded α this gives the correct order of magnitude for the quotients. For sums we think that the correct order is n2/(log n)a with some a <1, perhaps with 2 log 2 - 1, as a result of Pomerance and Sárközy suggests. We also give more general inequalities for sums, products and quotients formed with different sets. The proofs use geometric tools, mainly the Szemerédi-Trotter inequality.

AB - Let A be a set of n real numbers such that the number of distinct twofold sums is αn. We show that the number of twofold products is ≧ cn2/(α4 log n), and the number of quotients is ≧ cn2/min (α6, α 4 log n) with some absolute constant c. For bounded α this gives the correct order of magnitude for the quotients. For sums we think that the correct order is n2/(log n)a with some a <1, perhaps with 2 log 2 - 1, as a result of Pomerance and Sárközy suggests. We also give more general inequalities for sums, products and quotients formed with different sets. The proofs use geometric tools, mainly the Szemerédi-Trotter inequality.

KW - Product set

KW - Sumset

UR - http://www.scopus.com/inward/record.url?scp=30844446452&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=30844446452&partnerID=8YFLogxK

U2 - 10.1556/SScMath.40.2003.3.4

DO - 10.1556/SScMath.40.2003.3.4

M3 - Article

VL - 40

SP - 301

EP - 308

JO - Studia Scientiarum Mathematicarum Hungarica

JF - Studia Scientiarum Mathematicarum Hungarica

SN - 0081-6906

IS - 3

ER -