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 ${\hat{\lambda }}_n(\xi )$ 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 $n/2+\mathcal{O}(1)$.

34 pages, 1 figure, to appear in Transactions of the American Mathematical Society