Talk:Equidistributed sequence

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Sound

This sounds like the same thing as a uniform distribution. Can we just redirect this page there? —Caesura 12:38, 17 Dec 2004 (UTC)

NO! Uniform distribution is about random variables. This is about non-random sequences! (The first paragraph was incredibly confusing about just that point, but I've fixed it.) Michael Hardy 18:56, 10 October 2005 (UTC)[reply]

merge suggestion - go for it

Hi Mike,

I'm thinking that equidistribution mod 1 should be made into a sbsection of this article ... is taht what you have in mind? linas 23:54, 10 October 2005 (UTC)[reply]

I agree with this. Go ahead and do it. --12.222.158.49 05:53, 6 February 2006 (UTC) (Not mike.)[reply]

cleanup

After merge cleanup for redundancy, and expert attention. Ste4k 06:12, 15 July 2006 (UTC)[reply]

Discrepancy

The discrepancy needs to be defined in this article, and some work done conjointly on Discrepancy theory. Richard Pinch (talk) 07:11, 16 August 2008 (UTC)[reply]

I've made a start. Richard Pinch (talk) 10:49, 16 August 2008 (UTC)[reply]

Another merger

Should Equidistribution theorem be merged into this article? Richard Pinch (talk) 10:49, 16 August 2008 (UTC)[reply]

Explanation needed for |...| in the definition

Let (s₁, s₂, s₃, …) be a bounded sequence of real numbers. What is |{s₁, s₂, s₃, …}|?

TomyDuby (talk) 01:56, 24 September 2008 (UTC)[reply]

Do you mean

\{s_{1},\ldots ,s_{n}\}\cap [c.d]

? That's the number of elenments in the intersection of the first n elements of the sequence and the closed interval from c to d. Richard Pinch (talk) 06:33, 24 September 2008 (UTC)[reply]

Yes, I meant exactly that. Thanks for explanation!

TomyDuby (talk) 07:16, 24 September 2008 (UTC)[reply]

Problematic Example

I don't think that the following makes sense:

The sequence {αⁿ} is equidistributed mod 1 for almost all values of α.

Clearly all you need do is take any α with modulus less than 1 to see that you get a convergent sequence which doesn't have the equidistributed property? —Preceding unsigned comment added by Thudso (talk • contribs) 09:37, 26 August 2010 (UTC)[reply]

Almost all provided that α > 1. The result was proven by Koksma in 1935. Spacepotato (talk) 19:17, 28 August 2010 (UTC)[reply]

Cleaned up

Hi, I cleaned up this page (the headings were really confusing and unrelated - things related to equidistribution mod 1 were not under equidistribution mod 1, for instance) and added some content. Also, I decided to be bold and merge the article on Weyl's criterion into this page. This is my first serious edit in Wikipedia, any feedback is welcome :) Yoni (talk) 17:16, 21 February 2014 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Equidistributed sequence/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

I am wondering if the line beginning with "1.", in the "Properties" section

should read " {a[n]} is equidistributed on the interval [0,1]." rather than " {a[n]} is equidistributed modulo 1."

This is just a guess, but something seems wrong. Consider the sequence of all positive multiples of sqrt(2). Since sqrt(2) is irrational, the equidistribution theorem mentioned in the Examples section, states that this sequence is equidistributed modulo 1.

But then, applying the line starting with "2." under "Properties", and chosing f(x) = x, (i.e. the identity function) I compute the integral on the RHS to get 1/2, but for the LHS using multiples of sqrt(2)I get:

    lim (1/n) Sum( j * sqrt(2))  = lim (1/n) * sqrt(2) * n*(n+1)/2 = lim sqrt(2)/2 * (n+1) ---> infinity.

Since 1/2 does not equal infinity, something seems wrong.

Churchill17 (talk) 17:27, 22 May 2009 (UTC)[reply]

Last edited at 17:27, 22 May 2009 (UTC). Substituted at 14:34, 29 April 2016 (UTC)

"Weyl s equidistribution criterion" listed at Redirects for discussion

The redirect Weyl s equidistribution criterion has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 September 14 § Weyl s equidistribution criterion until a consensus is reached. 1234qwer 1234qwer 4 16:44, 14 September 2024 (UTC)[reply]