A thermodynamic classification of pairs of real numbers via the Triangle Multi-dimensional continued fraction

TL;DR

This paper introduces a novel classification scheme for pairs of real numbers based on the properties of the triangle map, a multi-dimensional continued fraction, inspired by thermodynamics and statistical mechanics.

Contribution

It generalizes previous continued fraction methods by incorporating thermodynamic concepts and geometric properties of the triangle map for classifying real number pairs.

Findings

New classification scheme for real number pairs

Utilizes properties of the triangle map

Connects number theory with thermodynamic ideas

Abstract

A new classification scheme for pairs of real numbers is given, generalizing earlier work of the author that used continued fraction, which in turn was motivated by ideas from statistical mechanics in general and work of Knauf and Fiala and Kleban in particular. Critical for this classification are the number theoretic and geometric properties of the triangle map, a type of multi-dimensional continued fraction.