We say that an $m$-tuple $\omega=(\omega_1,\ldots,\omega_m)$ satisfies *the Diophantine condition* of order $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ and integer $p_1,\ldots,p_m$ the inequality
$$\max\limits_{1 \leq j \leq m}|\omega_j q - p_{j}| \geq C \left(\frac{1}{q}\right)^{\frac{1+\nu}{m}}$$
is satisfied.

- Does the set of the Diophantine $m$-tuples has full measure?

For $m=1$ the set of numbers satisfying the Diophantine condition of order $\nu > 0$ is a set of full measure. This is due to a theorem of Khinchin that states if $f \colon \mathbb{N} \to \mathbb{R}_{+}$ and $\sum\limits_{q=1}^{\infty} f(q)$ converges then almost all numbers satisfy $|\omega q - p| \geq C f(q)$ for some $C=C(\omega)$.

- Is there a generalization of the Khinchin theorem for multi-dimensional situations?

I read somewhere that Khinchin has generalized this result, but no references were given.

I will be grateful for any help.