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Lonely Points in Simplices.

Maximilian Jaroschek1, Manuel Kauers2, Laura Kovács3

  • 1QAware Gmbh, Aschauer Straße 32, 81549 München, Germany.

Discrete & Computational Geometry
|January 6, 2023
PubMed
Summary
This summary is machine-generated.

Researchers identified "lonely" points within dilated standard simplices, which are points not equivalent to any other point in the set modulo a lattice. They explored conditions on the lattice (L) for an unbounded number of lonely points as dilation increases.

Keywords:
Discrete geometryInteger pointsLatticesPolytopes

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Area of Science:

  • Number Theory
  • Geometry
  • Computational Mathematics

Background:

  • The study investigates the concept of 'lonely' points within a set A, defined as points not equivalent modulo a lattice L to any other point in A.
  • The focus is on a specific set A, namely a dilated standard simplex in m-dimensional Euclidean space (R^m).

Purpose of the Study:

  • To identify lonely points for specific lattice (L) and dilated standard simplex (A) configurations.
  • To determine conditions on the lattice L that lead to an unbounded number of lonely points as the simplex dilation increases.

Main Methods:

  • The study involves analyzing the geometric and number-theoretic properties of points within a dilated simplex.
  • Methods likely include lattice point counting, geometric analysis of convex bodies, and modular arithmetic.

Main Results:

  • The research characterizes the conditions under which lonely points exist within the dilated simplex.
  • It establishes criteria for the lattice structure (L) that result in a growing number of lonely points with increasing simplex dilation.

Conclusions:

  • The findings provide insights into the distribution of points in dilated simplices relative to lattice structures.
  • Understanding these conditions is crucial for applications in areas like discrete geometry, coding theory, and crystallography.