On geometry of simultaneous approximation to three real numbers

TL;DR

This paper investigates the geometric structure of best approximations to three real numbers and establishes a sharper lower bound for the ratio of ordinary to uniform Diophantine approximation exponents, enhancing understanding of approximation quality.

Contribution

It introduces a new geometric perspective and derives an optimal lower bound for the ratio of exponents in simultaneous approximation to three real numbers.

Findings

Established a sharper lower bound for the ratio of exponents.

Provided a geometric framework for analyzing best approximations.

Achieved an optimal bound in terms of the geometry of approximation sequences.

Abstract

Considering simultaneous approximation to three numbers, we study the geometry of the sequence of best approximations. We provide a sharper lower bound for the ratio between ordinary and uniform exponent of Diophantine approximation, optimal in terms of this geometry.