A note on the relative growth of products of multiple partial quotients in the plane

Brown-Sarre, Adam ; Hussain, Mumtaz 

Canadian mathematical bulletin, Tome 66 (2023) no. 2, pp. 544-552

Résumé

Let $r=[a_1(r), a_2(r),\ldots ]$ 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 $n\geq 1$.

Détail

DOI : 10.4153/S0008439522000510

Mots-clés : Metric continued fractions, Hausdorff dimension, uniform Diophantine approximation