Diophantine exponents of lattices and growth of multidimensional analogues of partial quotients

TL;DR

This paper explores the relationship between Diophantine exponents of lattices and the growth of multidimensional continued fraction analogues, specifically Klein polyhedra, extending classical one-dimensional results to higher dimensions.

Contribution

It introduces a multidimensional analogue of continued fractions using Klein polyhedra and investigates the connection with Diophantine exponents in three dimensions.

Findings

Established a relation between Diophantine exponents and Klein polyhedra growth

Extended classical continued fraction theory to three dimensions

Provided new insights into multidimensional Diophantine approximation

Abstract

In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued fractions, we consider Klein polyhedra.