A devil's staircase from rotations and irrationality measures for Liouville numbers

TL;DR

This paper introduces a new function derived from Sturmian and Christoffel words that reveals the nature of irrationality measures and distinguishes algebraic integers from transcendental numbers through its continuity and differentiation properties.

Contribution

It constructs a novel increasing function from combinatorial words that encodes number-theoretic properties and differentiates types of real numbers based on irrationality measures.

Findings

Function is continuous at irrationals, discontinuous at rationals.

Assumes algebraic integers at rational points, transcendental at irrationals.

Differentiation of the function relates to irrationality measures.

Abstract

From Sturmian and Christoffel words we derive a strictly increasing function $\Delta:[0,\infty)\to\mathbb{R}$. This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of $\Delta$ distinguishes some irrationality measures of real numbers.