# Effective simultaneous rational approximation to pairs of real quadratic numbers

## TL;DR

This paper proves the existence of effective bounds for simultaneous approximation of pairs of quadratic real numbers from distinct quadratic fields, showing how closely they can be approximated by rationals.

## Contribution

It establishes effective, computable bounds for simultaneous rational approximation to pairs of quadratic numbers in different quadratic fields.

## Key findings

- Existence of positive real numbers  and c with effective bounds.
- For all sufficiently large q, max of distances  and  exceeds q^{-1 + au}.
- Results are explicitly computable and apply to pairs in distinct quadratic fields.

## Abstract

Let $\xi, \zeta$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $\tau$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q \xi \|, \|q \zeta\| \} > q^{-1 + \tau}, $$ where $\| \cdot \|$ denotes the distance to the nearest integer.