A geometrical approach to measure irrationality

TL;DR

This paper introduces a geometric method to quantify the irrationality of numbers through circular sector areas, linking it to continued fractions and analyzing asymptotic bounds for large sector radii.

Contribution

It presents a novel geometric framework connecting irrationality measures with continued fraction expansions and asymptotic analysis of sector areas.

Findings

Established bounds for sector areas as radius increases

Connected geometric measures with continued fraction properties

Analyzed asymptotic behavior of convergent denominators

Abstract

We present a geometric way of describing the irrationality of a number using the area of a circular sector $A(r)$. We establish a connection between this and the continued fraction expansion of the number, and prove bounds for $A(r)$ as $r\to\infty$ by describing the asymptotic behavior of the ratios of the denominators of the convergents.