On the Extremal Theory of Continued Fractions

Letting $$x=[a_1(x), a_2(x), \ldots ]$$ x = [ a 1 ( x ) , a 2 ( x ) , … ] denote the continued fraction expansion of an irrational number $$x\in (0, 1)$$ x ∈ ( 0 , 1 ) , Khinchin proved that $$S_n(x)=\sum \nolimits _{k=1}^n a_k(x) \sim \frac{1}{\log 2}n\log n$$ S n ( x ) = ∑ k = 1 n a k ( x ) ∼ 1 log 2 n log n in measure, but not for almost every $$x$$ x . Diamond and Vaaler showed that, removing the largest term from $$S_n(x)$$ S n ( x ) , the previous asymptotics will hold almost everywhere, this shows the crucial influence of the extreme terms of $$S_n (x)$$ S n ( x ) on the sum. In this paper we determine, for $$d_n\rightarrow \infty $$ d n → ∞ and $$d_n/n\rightarrow 0$$ d n / n → 0 , the precise asymptotics of the sum of the $$d_n$$ d n largest terms of $$S_n(x)$$ S n ( x ) and show that the sum of the remaining terms has an asymptotically Gaussian distribution.