Nearly k-Distance Sets

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.