# Equidistribution of set of numbers and moments

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.

• I think something like this should be true and it should follow from this: en.wikipedia.org/wiki/… Commented Jan 16, 2023 at 5:27
• @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? Commented Jan 16, 2023 at 5:54
• Loosely speaking. Formally we have Stone-Weierstrass: en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem Commented Jan 16, 2023 at 6:43