On the number of good rational approximations to algebraic numbers

Abstract:

We study rational approximations $x/y$ to algebraic and, more generally, to real numbers $\xi$. Given $\delta > 0$, and writing $L = \log (1 + \delta )$, the number of approximations with $|\xi - (x/y)| < {y^{ - 2 - \delta }}$ is $\leq {L^{ - 1}}\log \log H + {c_1}(\delta ,r)$ if $\xi$ is algebraic of degree $\leq r$ and of height $H$ , and is $\leq {L^{ - 1}}\log \log B + {c_2}(\delta )$ if $\xi$ is real and we restrict to approximations with $y \leq B$. It turns out that the dependency on $H$ resp. $B$ in these estimates is the best possible, i.e., that the summands ${L^{ - 1}}\log \log H$ resp. ${L^{ - 1}}\log \log B$ are optimal.