On the Extremal Theory of Continued Fractions
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چکیده
Letting x = [a1(x), a2(x), . . .] denote the continued fraction expansion of an irrational number x ∈ (0, 1), Khinchin proved that Sn(x) = ∑n k=1 ak(x) ∼ 1 log 2 n logn in measure, but not for almost every x. Diamond and Vaaler showed that removing the largest term from Sn(x), the previous asymptotics will hold almost everywhere, showing the crucial influence of the extreme terms of Sn(x) on the sum. In this paper we determine, for dn → ∞, dn/n → 0, the precise asymptotics of the sum of the dn largest terms of Sn(x) and show that the sum of the remaining terms has an asymptotically Gaussian distribution.
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تاریخ انتشار 2014