Abstract
AbstractLet
$r=[a_1(r), a_2(r),\ldots ]$
be the continued fraction expansion of a real number
$r\in \mathbb R$
. The growth properties of the products of consecutive partial quotients are tied up with the set admitting improvements to Dirichlet’s theorem. Let
$(t_1, \ldots , t_m)\in \mathbb R_+^m$
, and let
$\Psi :\mathbb {N}\rightarrow (1,\infty )$
be a function such that
$\Psi (n)\to \infty $
as
$n\to \infty $
. We calculate the Hausdorff dimension of the set of all
$ (x, y)\in [0,1)^2$
such that
$$ \begin{align*} \max\left\{\prod_{i=1}^ma_{n+i}^{t_i}(x), \prod_{i=1}^ma_{n+i}^{t_i}(y)\right\} \geq \Psi(n) \end{align*} $$
is satisfied for all
$n\geq 1$
.
Publisher
Canadian Mathematical Society