On a mixed problem in Diophantine approximation

TL;DR

This paper extends Diophantine approximation results to algebraic numbers of degree d, establishing the existence of infinitely many algebraic numbers close to a given algebraic number with bounds involving height and norm.

Contribution

It generalizes previous work by de Mathan and Teulié from degree 1 to arbitrary degree d, providing new bounds in mixed Diophantine approximation problems.

Findings

Existence of infinitely many algebraic numbers of degree d near a given algebraic number.

New bounds involving height and p-adic norm for approximation.

Extension of prior results to higher degrees.

Abstract

Let $d$ be a positive integer. Let $p$ be a prime number. Let $\alpha$ be a real algebraic number of degree $d+1$. We establish that there exist a positive constant $c$ and infinitely many algebraic numbers $\xi$ of degree $d$ such that $|\alpha - \xi| \cdot \min\{|\Norm(\xi)|_p,1\} < c H(\xi)^{-d-1} (\log 3 H(\xi))^{-1/d}$. Here, $H(\xi)$ and $\Norm(\xi)$ denote the na{\"\i}ve height of $\xi$ and its norm, respectively. This extends an earlier result of de Mathan and Teuli\'e that deals with the case $d=1$.