Continued fractions and parametric geometry of numbers

TL;DR

This paper explores the connection between continued fractions and the parametric geometry of numbers, revealing how the minima's behavior reflects the continued fraction expansion of a real number's distance to the nearest integer.

Contribution

It establishes a link between continued fractions and the parametric geometry of numbers for the case of a single real number.

Findings

Minima behavior mirrors continued fraction expansion

Connection between geometry of numbers and Diophantine approximation

Provides new insights into rational approximation of real numbers

Abstract

Recently, W. M. Schmidt and L. Summerer developed a new theory called Parametric Geometry of Numbers which approximates the behaviour of the successive minima of a family of convex bodies in $\mathbb{R}^{n}$ related to the problem of simultaneous rational approximation to given real numbers. In the case of one number, we show that the qualitative behaviour of the minima reflects the continued fraction expansion of the smallest distance from this number to an integer.