On simultaneous rational approximations to a real number, its square, and its cube

TL;DR

This paper establishes a new upper bound on how well transcendental real numbers can be simultaneously approximated by rational triples sharing the same denominator, refining previous bounds and impacting algebraic approximation theory.

Contribution

It introduces a refined upper bound on the uniform exponent of approximation for (xi, xi^2, xi^3), improving upon earlier results by Davenport and Schmidt.

Findings

New upper bound on uniform approximation exponent for (xi, xi^2, xi^3)

Sharper lower bound on approximation exponent by algebraic integers of degree ≤ 4

Refinement of previous approximation bounds for transcendental numbers

Abstract

We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of Davenport and Schmidt. As a consequence, we get a sharper lower bound on the exponent of approximation of such a number xi by algebraic integers of degree at most 4.