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Topic: An interesting set
Replies: 4   Last Post: Apr 20, 2013 9:12 AM

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 G. A. Edgar Posts: 2,510 Registered: 12/8/04
Re: An interesting set
Posted: Apr 20, 2013 9:12 AM

In article <k7b3n8pok9afdn2jbjpi3f9oid3263bb4t@4ax.com>,
<dullrich@sprynet.com> wrote:

> On Fri, 19 Apr 2013 08:10:55 -0700 (PDT), apoorv <skjshr@gmail.com>
> wrote:
>

> >I am not sure whether this set has been discussed earlier.Let the set
> >X contain alll binary decimals x, between 0 and 1 ,
> >such that every initial portion of the decimal Contains not more than
> >1/3 '1st'. For example , .00100... would be in the set.
> >Then as far as I can make out,X is uncountable and so is [0,1]-X. Both
> >are dense in each other.
> >What would be the measure of these sets?

>
> I'm pretty sure it's pretty clear that X has measure zero.
>
> Given N, let x(N) be the number of N-bit sequences with
> at most N/3 1's. Then it's not hard to show, from the
> Central Limit Theorem if not by something simpler,
> that x(N)/2^N -> 0 as N -> infinity.
>
>

Yes. Almost all reals in [0,1] have digits with limiting frequency 1/2
for digit 1. So Law of Large Numbers is enough, Central Limit Theorem
is overkill.

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

Date Subject Author
4/19/13 apoorv
4/19/13 apoorv
4/19/13 Butch Malahide
4/19/13 David C. Ullrich
4/20/13 G. A. Edgar