**Part 1. Equidistribution.**

As mentioned in the comments, the equidistribution theorem, states that in the limit any irrational value will produce an equidistributed sequence. That is, in the limit as $n \rightarrow \infty$, all values of $x$ are equally likely, and thus it is said that the range of the sequence is dense on the interval $[0,1]$. This was proven by Weyl, and so this theorem is often called Weyl's criterion.

However, as you have mentioned, equidistribution is quite a weak criterion and does not necessarily imply that for finite $n$ the sequence will fill the interval 'evenly' without large gaps or clusters. 

**Part 2. The Kronecker low discrepancy sequence in 1-dimension**

Sequences that satisfy this stronger criterion of 'evenness' are called [low discrepancy sequences][1]. There are many ways to construct low discrepancy sequences, including the van der Corput / Halton sequences, Niederreiter and Sobol sequences to name a few. However, by far the most simple to construct are the Kronecker (sometimes called Weyl) additive recurrence sequences, which are defined exactly as you describe in your question. That is, 
$$ x_n =  \lfloor \alpha n\rfloor =  \alpha n \; (\textrm{mod} \; 1), \quad n=1,2,3,... $$
 
It turns out that only certain values of $\alpha$ produce low discrepancy sequences. For number theoretic reasons, these special values of $\alpha$ are also correspond to numbers that are [badly approximable][2] (by the rationals). Intuitively one can say that the more badly approximable a number is by the rationals, the 'more irrational' it is. This is why the golden ratio, which is the most badly approximable number, is often described as the most irrational number.  The more irrational the value of $\alpha$ is, the better ('more even') the low discrepancy sequence will be. Thus, the golden ratio $\varphi  = \frac{1+\sqrt{5}}{2} $ produces the optimal low discrepancy sequence.  

Worth mentioning, is that any value of $\alpha$ related via the moebius transformation, 
$$ \alpha = \frac{a+b\varphi}{c+d\varphi} \quad \textrm{for integers} \;\;a,b,c,d \;\; \textrm{where} \;  |ad-bc|=1.$$
will also be optimal.

**Part 3. Kronecker low discrepancy sequences in higher dimensions**

Regarding the main part of your question as to if there is an equivalent process for higher dimensions, the answer is a resounding '**Yes!**'.

Some visualizations of various two dimensional low discrepancy sequences can be seen [here][3].

The $d$-dimensional Kronecker sequence is a natural extension of the 1-dimensional case. That is, for a given constant $\pmb{\alpha} = (\alpha_1, \alpha_2,...,\alpha_d)$, the infinite sequence of $d$-dimensional vectors $\pmb{x}_n \in U[0,1]^d$ defined as follows:

$$ \pmb{x}_n =  n\pmb{\alpha}  \; (\textrm{mod} \; 1), \quad n=1,2,3,... $$

is equidistributed if $\alpha_i$ are all independently irrational numbers. Furthermore, in many cases if $\alpha_i$ are the square roots of prime numbers, than the sequence will be a low discrepancy sequence. Unfortunately, it is non-trivial to know in advance if a particular set of prime numbers will result in a low discrepancy sequence.
For example, $\pmb{\alpha} = (\sqrt{2}, \sqrt{3})$ is not great, but as seen in the diagram above  $\pmb{\alpha} = (\sqrt{3}, \sqrt{7})$ is quite good.

Thankfully, in most applications, for moderately large $d$, the construction of such a $d$-dimensional low discrepancy sequence that is based on the square roots of the first $d$ prime numbers, produces quite good results. 

**Part 4. Generalizing the Golden ratio**

My blog post ["The unreasonable effectiveness of quasirandom sequences"][3], describes another method that produces significantly better and more reliable results. In this case, the $d$-dimensional constant vector $\pmb{\alpha}$ is based on a special value $\phi_d$ that is a $d$-dimensional generalisation of the golden ratio, $\phi = \frac{1+\sqrt{5}}{2}$.

That is, 
>> $$ \pmb{t}_n =  n \pmb{\phi}  \; (\textrm{mod} \; 1),  \quad n=1,2,3,... $$
>> where
>> $$ \pmb{\phi} =(\frac{1}{\phi_d^1}, \frac{1}{\phi_d^2},\frac{1}{\phi_d^3},...\frac{1}{\phi_d^d}), $$

>> $$ \textrm{and} \; \phi_d\ \textrm{is the smallest value of } \; x>0 \; \textrm{such that} $$

>> $$ x^{d+1}\;=x+1. $$

For $d=1$,  $ \phi_1 = 1.618033989... $, which is the canonical golden ratio.

For $d=2$, $ \phi_2 = 1.3247179572... $, which  is often called the [plastic constant][4] or silver ratio, and has some beautiful properties. This value was conjectured to most likely be the optimal value for a related two-dimensional problem [Hensley, 2002]. Jacob Rus has posted a beautiful visualization of this 2-dimensional low discrepancy sequence, which can be found [here][5].

For $d=3$, $ \phi_3 = 1.2207440846... $

**Summary**

Here is some Python code to create the $d$-dimensional generalization of the golden ratio-based Kronecker sequence that you cite in your question.

    # Use Newton-Rhapson-Method to calculate \phi_d
    def gamma(d):
        x=1.0000
        for i in range(20):
            x = x-(pow(x,d+1)-x-1)/((d+1)*pow(x,d)-1)
        return x

    d=3
    n=100

    g = gamma(d)
    alpha = np.zeros(d)                 
    for j in range(d):
        alpha[j] = pow(1/g,j+1) %1
    z = np.zeros((n, d))    
    seed = 0.5  #Optional seed value.
    for i in range(n):
        z = (seed + alpha*(i+1)) %1
    
    print(z)

Hope that helps!


  [1]: https://en.wikipedia.org/wiki/Low-discrepancy_sequence
  [2]: https://en.wikipedia.org/wiki/Diophantine_approximation
  [3]: http://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/
  [4]: https://en.wikipedia.org/wiki/Plastic_number
  [5]: https://arxiv.org/pdf/1603.08360.pdf