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