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

Question

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.

1. 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 $C=C(\omega)$.

2. 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.

Answer 1

I'm pretty sure that plenty of those kind of questions are covered in Cassels' book.

The modern approach to this kind of problems follows from dynamics on homogeneous spaces via Dani's correspondence, and in-particular this issue is tightly related to the question of divergence under the geodesic flow (or more generally, one-param. diagonalizable subgroup) of certain pieces of unipotent flows.

The main tool is the so-called quantitative non-divergence theorem of Kleinbock-Margulis (generalizing previous qualitative works of Margulis, Dani-Margulis) - "FLOWS ON HOMOGENEOUS SPACES AND DIOPHANTINE APPROXIMATION ON MANIFOLDS" https://arxiv.org/pdf/math/9810036.pdf

In particular your question easily follows from their extremely general Theorem A.

Edit - just to make the answer clear, the answer for 1 is Yes, and 2 is dealt in the Kleinbock-Margulis article (and many further developments, say by Kleinbock, Shah and others).

Answer 2

To answer 1) and 2) there is no need to use dynamics. In fact your question is about the convergence case of Khintchine's theorem, which is an easy consequence of Borel-Cantelli lemma.

The generalization for $m>1$ is indeed due to Khintchine. You can read about even further generalizations in Schmidt's book on Diophantine Approximation https://www.springer.com/gp/book/9783540097624 (the part called `measures'). Another nice reading is Dodson's survey paper, http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=1323&option_lang=eng. Hope it helps!