Simultaneous approximation of a real number by all conjugates of an algebraic number

TL;DR

This paper proves that the ratio 2/n is the best possible exponent for simultaneously approximating real irrationals using all conjugates of algebraic numbers of degree n, extending to algebraic integers of degree n+1.

Contribution

It establishes the optimal exponent for simultaneous approximation by conjugates of algebraic numbers and integers, using a method based on Davenport and Schmidt.

Findings

The ratio 2/n is the optimal approximation exponent for algebraic numbers of degree n.

The same exponent applies when approximating with all but one conjugate of algebraic integers of degree n+1.

The results extend the understanding of Diophantine approximation for algebraic conjugates.

Abstract

Using a method of H. Davenport and W. M. Schmidt, we show that, for each positive integer n, the ratio 2/n is the optimal exponent of simultaneous approximation to real irrational numbers 1) by all conjugates of algebraic numbers of degree n, and 2) by all but one conjugates of algebraic integers of degree n+1.