On distinct sums and distinct distances

Research output: Contribution to journalArticle

22 Citations (Scopus)

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 (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.

Original languageEnglish
Pages (from-to)275-289
Number of pages15
JournalAdvances in Mathematics
Volume180
Issue number1
DOIs
Publication statusPublished - Dec 1 2003

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Distinct
Natural logarithm
Discrete Geometry
Several Variables
Erdös
Pairwise
Disjoint
Random variable
Entropy
Infinity
Tend
Arbitrary

Keywords

  • Distinct distances
  • Entropy
  • Erdos problems

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Advances in Mathematics, Vol. 180, No. 1, 01.12.2003, p. 275-289.

Research output: Contribution to journalArticle

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