On geometry of numbers and uniform rational approximation to the Veronese curve

TL;DR

This paper investigates the geometry of numbers and uniform rational approximation to the Veronese curve, providing bounds on approximation exponents and slight improvements on classical bounds using advanced number theory techniques.

Contribution

It combines recent results with parametric geometry of numbers to bound dual linear form exponents and improves upper bounds for the classical exponent of uniform Diophantine approximation.

Findings

Bound the dual linear form exponents in rational approximation to the Veronese curve.

Slightly improve the upper bound for the classical exponent of uniform Diophantine approximation for even n ≥ 4.

Establish stronger results for numbers nearly satisfying the equality in Marnat and Moshchevitin's estimate.

Abstract

Consider the classical problem of rational simultaneous approximation to a point in $\mathbb{R}^{n}$. The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and Moshchevitin in 2018. Recently Nguyen, Poels and Roy provided information on the best approximating rational vectors to points where the gap is close to this minimal value. Combining the latter result with parametric geometry of numbers, we effectively bound the dual linear form exponents in the described situation. As an application, we slightly improve the upper bound for the classical exponent of uniform Diophantine approximation $\widehat{\lambda}_{n}(\xi)$, for even $n\geq 4$. Unfortunately our improvements are small, for $n=4$ only in the fifth decimal digit. However, the underlying method in principle can be improved with more effort to provide better bounds. We indeed establish reasonably stronger results for numbers which almost satisfy equality in the estimate by Marnat and Moshchevitin. We conclude with consequences on the classical problem of approximation to real numbers by algebraic numbers/integers of uniformly bounded degree.