The distribution of the largest digit in continued fraction expansions

Let [ a 1 ( x ), a 2 ( x ), . . .] be the continued fraction expansion of x ∈ [0,1). Write T n ( x )=max{ a k ( x ):1 ≤ k ≤ n }. Philipp [ 6 ] proved that Okano [ 5 ] showed that for any k ≥ 2, there exists x ∈ [0, 1) such that T ( x )=1/log k . In this paper we show that, for any α ≥ 0, the set is of Hausdorff dimension 1.