A note on two linear forms

TL;DR

This paper investigates how well real numbers can be approximated by quadratic algebraic numbers, using the uniform Diophantine exponent of a specific quadratic linear form, contributing to number theory and Diophantine approximation.

Contribution

It provides a new result relating the approximation of real numbers by quadratic algebraic numbers to the uniform Diophantine exponent of a related linear form.

Findings

Establishes bounds on approximation quality based on the uniform Diophantine exponent.

Connects approximation properties to the structure of quadratic forms.

Advances understanding of Diophantine approximation for quadratic algebraic numbers.

Abstract

We prove a result on approximations to a real number $\theta$ by algebraic numbers of degree $\le 2$ in the case when we have information about the uniform Diophantine exponent $\hat{\omega}$ for the linear form $x_0 +\theta x_1+\theta^2x_2$.