# Need information on equidistribution

1. Dec 28, 2011

### ArcanaNoir

1. The problem statement, all variables and given/known data

I'm working on some proofs involving equivalent definitions of an equidistributed sequence. I need some resources to learn about equidistribution. Any links or book titles would be greatly appreciated.

2. Relevant equations

For an equidistributed sequence,
$$\lim_{n\to \infty } \frac{f(x_1)+f(x_2)+...+f(x_n)}{n}=\int^1_0 f(x) \, \mathrm{d} x$$

Also, the probability of choosing an $x_i$ from any subinterval of a sequence equidistributed over [0,1] is equal to the length of the subinterval.

3. The attempt at a solution

I've looked up equidistribution on google but I'm not finding enough explanation or detail. I also checked the forum for the term equidistribution and reviewed the results.

2. Dec 28, 2011

### rhody

Hi Arcana,

I ran a google scholar search on: equidistribution theorem and got this.

The "Cited by xxx" in the lower left hand corner of the search results indicates that
some links/documents are more valued than others. I hope this gives you a start.

Rhody...

3. Dec 28, 2011

### I like Serena

Hi Arcana!

It looks a bit as if you are studying on Riemann integrals:
http://en.wikipedia.org/wiki/Riemann_integral
which is the definition of the standard integral.

A Riemann integral is defined by using a partition of the interval and calculating the Riemann sum of this partition.
Loosely speaking, the Riemann integral is the limit of the Riemann sums as the partitions get finer.

In particular you can choose a sequence of sub intervals that all have the same size, giving you an equipartition.
If you pick an equidistribution with a random value from each interval, you will still get the Riemann integral (by definition).