Lonely Points in Simplices

TL;DR

This paper investigates the existence and quantity of 'lonely points' in dilated simplices within lattice structures, providing conditions under which the number of such points becomes unbounded as the simplex expands.

Contribution

It introduces a framework for identifying lonely points in simplices relative to lattice equivalence and establishes conditions for their unbounded growth.

Findings

Characterization of lonely points in specific lattice-simplex configurations

Conditions under which the number of lonely points grows without bound

Insights into lattice equivalence classes in geometric structures

Abstract

Given a lattice L in Z^m and a subset A of R^m, we say that a point in A is lonely if it is not equivalent modulo L to another point of A. We are interested in identifying lonely points for specific choices of L when A is a dilated standard simplex, and in conditions on L which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.