Simultaneous rational approximation to successive powers of a real number

Abstract

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