Simultaneous rational approximation to successive powers of a real number

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 $n/2+\mathcal {O}(1)$.