The distribution of the large partial quotients in continued fraction expansions

Abstract

The existence of large partial quotients destroys many limit theorems in the metric theory of continued fractions. To achieve some variant forms of limit theorems, a common approach mostly used in practice is to discard the largest partial quotient, while this approach works in obtaining limit theorems only when there cannot exist two terms of large partial quotients in a metric sense. Motivated by this, we are led to consider the metric theory of points with at least two large partial quotients. More precisely, denoting by [a₁(x), a₂(x),…] the continued fraction expansion of x ∈ [0, 1) and letting ψ: ℕ → ℝ⁺ be a positive function tending to infinity as n → ∞, we present a complete characterization on the metric properties of the set, i.e.,

$$E\left( \psi \right) = \left\{ {x \in \left[ {0,1} \right):\exists \,1 \leqslant k \ne \ell \leqslant n,\,\,{a_k}\left( x \right) \geqslant \psi \left( n \right),\,\,{a_\ell }\left( x \right) \geqslant \psi \left( n \right)\,\,{\rm{for}}\,{\rm{infinitely}}\,{\rm{many}}\,n \in \mathbb{N}} \right\}$$

in the sense of the Lebesgue measure (the Borel-Bernstein type result) and the Hausdorff dimension (the Jarnik type result). The main result implies that any finite deletion from a₁(x) + ⋯ + a_n(x) cannot result in a law of large numbers.