Discretizations of isometries

TL;DR

This paper studies how repeatedly discretizing isometries like rotations affects the density of points in integer lattices, showing that this density diminishes to zero over time, leading to information loss.

Contribution

It demonstrates that successive discretizations of generic isometries cause the lattice point density to tend to zero, revealing limitations of naive numerical rotation algorithms.

Findings

Density of lattice points tends to zero over time

Naive discretization algorithms lead to information loss

Reveals limitations in numerical implementations of rotations

Abstract

This paper deals with the dynamics of discretizations of isometries of $\mathbf R^n$, and more precisely the density of the successive images of $\mathbf Z^n$ by the discretizations of a generic sequence of isometries. We show that this density tends to 0 as the time goes to infinity. Thus, in general, all the information of a numerical image will be lost by applying many times a naive algorithm of rotation.