Let $\omega=(\omega_1,\ldots,\omega_{m})$ be an $m$-tuple of real numbers. Let $|\omega|_{m}:=\sup\limits_{1 \leq j \leq m}|\omega_j|_{1}$ be a metric on flat torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\mathbb{Z}^m$, i. e. $|\theta|_{1}$ is the distance from $\theta$ to nearest integer.

We say that an $m$-tuple $\omega$ satisfies *the Diophantine condition of order* $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ the inequality
$$|\omega q|_{m} \geq C \left(\frac{1}{q}\right)^{\frac{1+\nu}{m}}$$
is satisfied.

Suppose we have some algorithm that provides a sequence of convergents (fractions which approximate $\omega$) with denominators $\{ q_{k} \}$. I want to find such an algorithm, which satisfyes the following properties

**Property 1.** If $\omega$ satisfies the Diophantine condition, then $q_{k+1}=O\left(q^{1+\nu}_{k}\right)$.

**Property 2.** For many (in some sense) $\omega$ there is a constant $\hat{C}>0$ such that the estimate
$$\sum\limits_{k=N}^{\infty}\frac{1}{q_{k}} \leq \hat{C}\frac{1}{q_{N}}$$
holds for all large enough $N$.

The classic continued fraction algorithm for $m=1$ provides the required denominators: property 1 is equivalent to the Diophantine condition and Property 2 holds for all real numbers.

Are there known multi-dimensional algorithms, satisfying the above properties? I will be grateful for any help.