Approximation of complex algebraic numbers by algebraic numbers of bounded degree

TL;DR

This paper explores the approximation quality of complex algebraic numbers by algebraic numbers and integers of bounded degree, revealing the existence of numbers with exceptional approximation properties.

Contribution

It demonstrates the existence of complex algebraic numbers with degrees greater than n that are better approximable by degree n algebraic numbers, and worse approximable by degree n+1 algebraic integers.

Findings

Existence of algebraic numbers with superior approximation by degree n algebraic numbers.

Such numbers are more badly approximable by degree n+1 algebraic integers.

Results highlight nuanced approximation behaviors depending on algebraic degree.

Abstract

We investigate how well complex algebraic numbers can be approximated by algebraic numbers of degree at most n. We also investigate how well complex algebraic numbers can be approximated by algebraic integers of degree at most n+1. It follows from our investigations that for every positive integer n there are complex algebraic numbers of degree larger than n that are better approximable by algebraic numbers of degree at most n than almost all complex numbers. As it turns out, these numbers are more badly approximable by algebraic integers of degree at most n+1 than almost all complex numbers.