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arXiv · 2109.14141

Simultaneous rational approximation to successive powers of a real number

Anthony Poëls·Damien Roy
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Abstract

We develop new tools leading, for each integer $n\ge 4$, to a significantly improved upper bound for the uniform exponent of rational approximation $\widehat{\lambda}_n(\xi)$ to successive powers $1,\xi,\dots,\xi^n$ of a given real transcendental number $\xi$. As an application, we obtain a refined lower bound for the exponent of approximation to $\xi$ by algebraic integers of degree at most $n+1$. The new lower bound is $n/2+a\sqrt{n}+4/3$ with $a=(1-\log(2))/2\simeq 0.153$, instead of the current $n/2+\mathcal{O}(1)$.

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Anthony Poëls, Damien Roy. 2021-09-29. Simultaneous rational approximation to successive powers of a real number. https://arxiv.org/abs/2109.14141

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