Lonely Points in Simplices

Abstract

Abstract Given a lattice $$L\subseteq \mathbb Z^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mi>Z</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> </mml:math> and a subset $$A\subseteq \mathbb R^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> </mml:math> , we say that a point in A is lonely if it is not equivalent modulo $$L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> to another point of A . We are interested in identifying lonely points for specific choices of $$L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> when A is a dilated standard simplex, and in conditions on $$L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.

Locations

• Discrete & Computational Geometry - View - PDF
• PubMed Central - View
• PubMed - View