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Equidistribution of a set of numbers $\{s_1, s_2, s_3, \cdots\}$, loosely speaking, is the property that the proportion of terms falling in a subinterval is proportional to the length of that subinterval (a more precise definition can be found in Wikipedia).

For simplicity let's restrict to the case of $s_n$ being real numbers in the interval $[0,1]$; then an example of an equidistributed sequence is $s_n = n\alpha \mod 1$ where $\alpha$ is an irrational number.

Is equidistribution equivalent to showing that the moments of the set of numbers match the moments of a uniform random variable $X$ on the interval, for all moments?

E.g. in the case of the interval $[0,1]$ can I say $\{s_n\}$ is equidistributed if $$ \lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^n s_n^k = \mathbb{E}[X^k]=\int_0^1dx x^k=\frac{1}{k+1} $$ for all non-negative integer $k$?

Why or why not? Thanks.

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  • $\begingroup$ I think something like this should be true and it should follow from this: en.wikipedia.org/wiki/… $\endgroup$ Commented Jan 16, 2023 at 5:27
  • $\begingroup$ @QiaochuYuan Thanks. Are you using something like that the monomials $x^k$ form a basis for Riemann integrable functions (is that even true haha) hence the equivalence with what you linked? $\endgroup$
    – nervxxx
    Commented Jan 16, 2023 at 5:54
  • $\begingroup$ Loosely speaking. Formally we have Stone-Weierstrass: en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem $\endgroup$ Commented Jan 16, 2023 at 6:43

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