### Abstract

The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all (_{2}^{s})n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least n^{ds}, where d_{s} = 1/c_{⌈s/2⌉} is defined in the paper and tends to e^{-1} as s goes to infinity. Here e is the base of the natural logarithm. As an application we improve the Solymosi-Tóth bound on an old Erdos problem: we prove that n distinct points in the plane determine Ω(4e/n^{5e-1-E}) distinct distances, where E > 0 is arbitrary. Our bound also finds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables.

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

Pages (from-to) | 275-289 |

Number of pages | 15 |

Journal | Advances in Mathematics |

Volume | 180 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1 2003 |

### Fingerprint

### Keywords

- Distinct distances
- Entropy
- Erdos problems

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*180*(1), 275-289. https://doi.org/10.1016/S0001-8708(03)00004-5

**On distinct sums and distinct distances.** / Tardos, G.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 180, no. 1, pp. 275-289. https://doi.org/10.1016/S0001-8708(03)00004-5

}

TY - JOUR

T1 - On distinct sums and distinct distances

AU - Tardos, G.

PY - 2003/12/1

Y1 - 2003/12/1

N2 - The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all (2s)n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least nds, where ds = 1/c⌈s/2⌉ is defined in the paper and tends to e-1 as s goes to infinity. Here e is the base of the natural logarithm. As an application we improve the Solymosi-Tóth bound on an old Erdos problem: we prove that n distinct points in the plane determine Ω(4e/n5e-1-E) distinct distances, where E > 0 is arbitrary. Our bound also finds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables.

AB - The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all (2s)n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least nds, where ds = 1/c⌈s/2⌉ is defined in the paper and tends to e-1 as s goes to infinity. Here e is the base of the natural logarithm. As an application we improve the Solymosi-Tóth bound on an old Erdos problem: we prove that n distinct points in the plane determine Ω(4e/n5e-1-E) distinct distances, where E > 0 is arbitrary. Our bound also finds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables.

KW - Distinct distances

KW - Entropy

KW - Erdos problems

UR - http://www.scopus.com/inward/record.url?scp=0346269038&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0346269038&partnerID=8YFLogxK

U2 - 10.1016/S0001-8708(03)00004-5

DO - 10.1016/S0001-8708(03)00004-5

M3 - Article

VL - 180

SP - 275

EP - 289

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 1

ER -