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

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.

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References in corpus (1)

• An irrationality measure for Liouville numbers and conditional measures for Euler's constant