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As mentioned in the comments, the equidistribution theorem states that any irrational value will produce an equidistributed sequence. That is, in the limit as $n \rightarrow \infty$, all finite intervals are equally likely.

As mentioned in the comments, the equidistribution theorem states that any irrational value will produce an equidistributed sequence. That is, in the limit as $n \rightarrow \infty$, all finite subintervals of $(0,1)$ are equally likely.

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.

As mentioned in the comments, the equidistribution theorem states that any irrational value will produce an equidistributed sequence. That is, in the limit as $n \rightarrow \infty$, all finite intervals are equally likely.

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

For $d=2$, $ \phi_2 = 1.3247179572... $, which  is often called the plastic constant 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.

Two things worth mentioning. First, 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.

And secondly, $\alpha = 1+\sqrt{2} $ turns out to be the next most badly approximable number -- and therefore the next most optimal value for the construction of a low discrepancy sequence.

Springborn as well as Spalding give number theoretic reasons why this is the second most badly approximable number, and also why $\alpha = \frac{1}{10} (9+\sqrt{221}$ is the third-most badly approximable number.

Interestingly the continued fraction for these three values are: $$ \varphi = 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}} = [1,1,1,1,...]$$ $$ 1+\sqrt{2} = 2+\frac{1}{2+\frac{1}{2+\frac{1}{2+...}}} = [2,2,2,2,2,2] $$ $$ \frac{1}{10}(9+\sqrt{221}) = 2+\frac{1}{2+\frac{1}{1+\frac{1}{1+...}}} = [2,2,1,1,2,2,1,1,2,2,1,1,..] $$

Part 3. Kronecker low discrepancy sequences in higher dimensions

For $d=2$, $ \phi_2 = 1.3247179572... $, which  is often called the plastic constant 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 interactive/dynamic visualization of this 2-dimensional low discrepancy sequence, which can be found here.