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Nearly k-Distance Sets.

Nóra Frankl1,2, Andrey Kupavskii3,4

  • 1School of Mathematics and Statistics, The Open University, Milton Keynes, UK.

Discrete & Computational Geometry
|October 9, 2023
PubMed
Summary
This summary is machine-generated.

This study investigates k-distance sets in Euclidean spaces, finding that the number of distinct distances between points grows with dimension. This answers a long-standing question in combinatorial geometry.

Keywords:
Erdős distance problemsTurán-type problemsk-Distance sets

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Area of Science:

  • Combinatorial Geometry
  • Discrete Geometry
  • Geometric Set Systems

Background:

  • Introduces the concept of k-distance sets, where distances between any two points in a set S are limited to k values.
  • Highlights the classical problem of determining the maximum size of a k-distance set in d-dimensional Euclidean space (ℝd).

Purpose of the Study:

  • To study the quantity N(S) and its relationship with the maximum size of a k-distance set in ℝd.
  • To address a Turán-type problem concerning the number of pairs of points with specific distances, given constraints on minimum separation.

Main Methods:

  • Analysis of the quantity N(S) related to k-distance sets.
  • Investigating Turán-type problems in geometric settings.
  • Establishing connections between different geometric quantities.

Main Results:

  • Proves that N(S) is proportional to d for large d and fixed k.
  • Demonstrates that for fixed k and sufficiently large d, N(S) is proportional to d.
  • Provides an exact answer to a Turán-type problem for specific ranges of k and d.

Conclusions:

  • The results contribute to understanding the distribution and properties of distances within point sets in Euclidean spaces.
  • Answers a question posed by Erdős, Makai, and Pach regarding k-distance sets.
  • Extends previous work on Turán-type problems in discrete geometry.