Let $a_1, \cdots, a_k \in [0, 1)$ be real numbers such that $1, a_1, \cdots, a_k$ are independent over the rational numbers. By the Weyl equidistribution criterion in $k$-dimensions, we know that the sequence of points $\{(a_1 n, \cdots, a_kn)\}_{n=1}^\infty$ is uniformly distributed mod $1$. This amounts to saying that for any $k$-cell $\prod_{j=1}^k [c_j, d_j) \subset [0, 1)^k$, we have $$\lim_{N \rightarrow \infty} \frac1N\#\{n \in \mathbb N: n \le N, (\{a_1 n\}, \cdots, \{a_kn\}) \in \prod_{j=1}^k [c_j, d_j)\} = \prod_{j=1}^k (d_j - c_j),$$ where $\{y\}$ denotes the fractional part of a real number $y$. I have often heard that the functions $X_j(t) := a_j t$ mod $1$ can be viewed as "approximately independent random variables" taking values in $[0, 1)$, and have been trying to make this notion precise in a probabilistic setting.
What I have tried is the following: I let $T>0$ be some large real number and considered the probability measure $\mathbb P$ on $[0, T]$ given by $\mathbb P(A) = \frac1T \lambda(A)$ with $\lambda(A)$ denoting the Lebesgue measure of any (Lebesgue measurable) set $A \subset [0, T]$. Then, I defined the random variables $X_j: [0, T] \rightarrow \mathbb R$ given by $X_j(t):= \{a_j t\}$, whose cumulative distribution functions can be calculated to be $$\mathbb P(X_j \le x) = \begin{cases} 0, &\text{ if } x \le 0\\ \frac{\lfloor a_j T \rfloor + 1}{a_j T}x, &\text{ if } 0 < x \le \{a_j T\}\\ \frac{\lfloor a_jT \rfloor}{a_j T}x + 1 - \frac{\lfloor a_j T \rfloor}{a_j T}, &\text{ if } \{a_j T\} < x \le 1\\ 1, &\text{ if } x \geq 1\\ \end{cases}$$ To have the random variables $X_j$ behave approximately independently, I should be able to show that the difference between the joint distribution $\mathbb P(X_1 \le x_1, \cdots, X_k \le x_k)$ and the product of the distributions $\prod_{j=1}^k \mathbb P(X_j \le x_j)$ is small; perhaps $o(1)$ as $T \rightarrow \infty$, uniformly for any choice of $x_1, \cdots, x_k \in (0, 1)$. It would be even better if I can bound the difference between the joint probability density and the product of the individual probability densities (at points of differentiability).
However, I have been unable to measure or bound either of these differences in my setting, since any of my attempts (so far) to explicitly compute joint densities has become messy. Which also makes me doubt whether this intuition I have of the $X_j$'s being "approximately independent" is correct in the first place.
Hence, I would really appreciate any thoughts, suggestions or references on this kind of probabilistic treatment (I have not found any reference so far either). Sorry if this question is naive, I am not very comfortable with probability or measure theory.
