A generalization of Thue's theorem to packings of non-equal discs, and an application to a discrete approximation of entropy

TL;DR

This paper extends Thue's theorem to packings of non-equal discs with a specific non-overlapping condition, establishing an upper bound on packing density and applying it to approximate the entropy functional in 2D.

Contribution

It generalizes Thue's theorem to non-uniform disc packings under a new inflation condition and connects this to entropy approximation.

Findings

Maximum packing density is at most π/(2√3).

The result applies to non-equal disc packings with a specific inflation property.

Provides a discrete approximation method for the 2D entropy functional.

Abstract

In this paper we generalize the classical theorem of Thue about the optimal circular disc packing in the plane. We are given a family of circular discs, not necessarily of equal radii, with the property that the inflation of every disc by a factor of $2$ around its center does not contain any center of another disc in the family (notice that this implies that the family of discs is a packing). We show that in this case the density of the given packing is at most $\frac{\pi}{2\sqrt{3}}$, which is the density of the optimal unit disc packing. This result is used to obtain a discrete approximation to the Entropy functional in two dimensional domain.