Testing the transcendence conjectures of a modular involution of the real line and its continued fraction statistics

TL;DR

This paper investigates the properties of a new involution J on the real line, testing conjectures about its algebraic and transcendental values, and analyzing its continued fraction statistics through theoretical and experimental methods.

Contribution

It introduces and studies the involution J, providing evidence for conjectures on algebraic to transcendental number transformation and analyzing its continued fraction behavior.

Findings

Support for the conjecture that J maps algebraic numbers of degree ≥3 to transcendental numbers

Theoretical results on continued fraction statistics of J's generic values

Comparison between experimental and theoretical continued fraction data

Abstract

We study the values of the recently introduced involution J (jimm) of the real line, which is equivariant with the action of the group PGL(2,Z). We test our conjecture that this involution sends algebraic numbers of degree at least three to transcendental values. We also deduce some theoretical results concerning the continued fraction statistics of the generic values of this involution and compare them with the experimental results.