Simultaneous approximation to a real number and to its cube

TL;DR

This paper investigates the limits of how well real numbers and their cubes can be simultaneously approximated by rationals, establishing a new upper bound for the approximation exponent in the case of triples involving a cube.

Contribution

It introduces a new upper bound for the uniform exponent of simultaneous approximation to (1,x,x^3) for Q-linearly independent triples, extending previous results for quadratic cases.

Findings

New upper bound for approximation exponent: approximately 0.7038.

Properties of minimal points sequence for smaller exponents.

Comparison with known bounds for quadratic approximations.

Abstract

It is known that, for each real number x such that 1,x,x^2 are linearly independent over Q, the uniform exponent of simultaneous approximation to (1,x,x^2) by rational numbers is at most (sqrt{5}-1)/2 (approximately 0.618) and that this upper bound is best possible. In this paper, we study the analogous problem for Q-linearly independent triples (1,x,x^3), and show that, for these, the uniform exponent of simultaneous approximation by rational numbers is at most 2(9+sqrt{11})/35 (approximately 0.7038). We also establish general properties of the sequence of minimal points attached to such triples that are valid for smaller values of the exponent.