I have a question about equidistribution. In Wikipedia, Equidistributed sequence shows in the equation in the definition that $n$ reaches almost infinity.
I was just wanting to make sure if equidistribution worked for much smaller sequences, like that of only $100$ or so numbers or something like that. It states a sequence of numbers, but without any specific characteristics. And I assume that for the equation to be true $n$ has to reach near infinity, or else it will only work for certain instances and not all.
Please correct me if I am wrong. Thanks!
From the definition on wikipedia: A sequence ${s_1, s_2, s_3 ...}$, $s_i \in \mathbb{R}$ is said to be ''equidistributed'' on an interval $[a,b]$ if for any subinterval $[c,d]$ of $[a,b]$ we have:
$\lim_{n\to\infty}{ \left|\{\,s_1,\dots,s_n \,\} \cap [c,d] \right| \over n}={d-c \over b-a} . \,$
Your question is about a finite sequence (i.e., one in which the $s_i$ are not defined for $i$ bigger than some $N$). The problem is then that it is easy to find a non empty subinterval which does not contain any of the $s_i$ (e.g. take $[0, \frac{\text{min}\{s_i\}}{2}]$, if $0$ is not in the sequence).
For any of these intervals, the left hand side of the equation above will equal 0, but the right hand side is clearly $>0$, so finite sequences cannot be "equidistributed".