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HAUSDORFF DIMENSION FOR THE SET OF POINTS CONNECTED WITH THE GENERALIZED JARNÍK–BESICOVITCH SET
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 112 / Issue 1 / February 2022
- Published online by Cambridge University Press:
- 07 December 2020, pp. 1-29
- Print publication:
- February 2022
-
- Article
- Export citation
-
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.
