Distances on a one-dimensional lattice from noncommutative geometry

TL;DR

This paper analytically derives exact formulas for distances on a one-dimensional lattice within a noncommutative geometry framework, revealing a specific averaging property for distances to even points.

Contribution

It provides the first analytical proof of exact distance formulas on a 1D lattice in noncommutative geometry, extending prior numerical or heuristic approaches.

Findings

Exact distance formulas derived analytically

Distance to even points is the geometric mean of neighboring odd points' distances

Extends previous work by Bimonte-Lizzi-Sparano

Abstract

In the following paper we continue the work of Bimonte-Lizzi-Sparano on distances on a one dimensional lattice. We succeed in proving analytically the exact formulae for such distances. We find that the distance to an even point on the lattice is the geometrical average of the ``predecessor'' and ``successor'' distances to the neighbouring odd points.