The Continuants of Transcendental Numbers in Continued Fraction Expansion

In this paper, we want to explain that if a purely transcendental set is determined merely by the properties of the individual continuants, besides algebraic numbers, most transcendental numbers are excluded from this set. Namely, let φ be a positive function defined on N and set

A (φ) = { x ∈ [ :1,0 ) q n (x ) ≥ φ ( n ) , infinitely many n }.

If A (φ ) is a purely transcendental set, then the set A (φ ) is of Hausdorff dimension at most one-half.

Let q n( θ ) be n-th continuant of θ in its continued fraction expansion.For any irrational θ ∈ [0,1) . Davenport and Roth showed that if θ satisfies

loglogqn>cn/√logn for, infinitely many n ∈ N , for all c > ,0 then θ must be transcendental. We say a set A purely transcendental set, if all the elements in A are transcendental.