In this article we calculate the Hausdorff dimension of the seta n+1 (x)a n (x) ≥ Φ(n) for infinitely many n ∈ N and a n+1 (x) < Φ(n) for all sufficiently large n ∈ N where Φ : N → (1, ∞) is any function with lim n→∞ Φ(n) = ∞. This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improvable numbers.
In this article we aim to investigate the Hausdorff dimension of the set of points
$x \in [0,1)$
such that for any
$r\in \mathbb {N}$
,
$$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} \end{align*} $$
holds for infinitely many
$n\in \mathbb {N}$
, where h and
$\tau $
are positive continuous functions, T is the Gauss map and
$a_{n}(x)$
denotes the nth partial quotient of x in its continued fraction expansion. By appropriate choices of
$r,\tau (x)$
and
$h(x)$
we obtain various classical results including the famous Jarník–Besicovitch theorem.
The classical Khintchine and Jarník theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grows with a certain rate. Recently it was observed that the growth of product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet's theorem. In this paper we consider the products of several consecutive partial quotients raised to different powers. Namely, we find the Lebesgue measure and the Hausdorff dimension of the following set:where t i ∈ R + for all 0 ≤ i ≤ m − 1, and Ψ : N → R ≥1 is a positive function.
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