Re: Pi, randomness of numbers?

From: r.e.s. (r.s_at_ZZmindspring.com)
Date: 12/08/04 

Date: Wed, 08 Dec 2004 07:56:17 GMT

"Phil Carmody" <thefatphil_demunged@yahoo.co.uk> wrote...
> "r.e.s." <r.s@ZZmindspring.com> writes:
>> "Phil Carmody" <thefatphil_demunged@yahoo.co.uk> wrote ...
>> > "r.e.s." <r.s@ZZmindspring.com> writes:
>> >> "Phil Carmody" <thefatphil_demunged@yahoo.co.uk> wrote ...

>> >> > Except for the asymptotically negligible deviations proved by
>> >> > Mahler, that is.
>> >>
>> >> That appears not to be an exception, since the behaviour described
>> >> by Mahler's theorem is just what one would expect of a real number
>> >> chosen at random in the interval (3,4).
>> >>
>> >> For details, see http://tinyurl.com/6gl4y.
>> >
>> > But the behaviour of Pi was not being compared with other arbitrary
>> > transcendentals, it was being compared with mythical "normal"
>> > behaviour.
>>
>> The poster asks about 'randomness'. With respect to Mahler's
>> theorem, the digits of pi exhibit a behaviour which is just
>> as expected for a number formed by adjoining random iid digits.
>> (The probability of it behaving otherwise is less than ~10^-12.)
>
> I don't believe that, as stated, to be true.

Let S(x) stand for the following statement about a real number x:

       For all integers p,q > 1, |x - p/q| > 1/q^42.

Then Mahler's theorem is S(pi), while the cited proof shows that
if X is uniformly distributed on interval (3,4), then

       pr( S(X) ) > 1 - 2(Zeta(42)-1) > 1 - 10^-12

as stated above.

> An arbitrary transcendental will have a Mahlerian deviation with
> regard to distribution of digit-sequences.

I doubt it -- S(x) might not be true for *all* transcendentals x;
but the theorem shows that it's true for at least an overwhelming
proportion of the transcendentals (and of all the reals) in (3,4).

> I don't believe by concatenating sequences of IID digits you are
> constructing an arbitrary transcendental, precisle because such a
> sequence cannot have the Mahlerian deviation.

The random number X = 3 . d_1 d_2 d_3 ... (using decimal notation)
is uniform on the interval (3,4) if the d_i are iid uniform on
{0,1,...,9}. And if X is uniform on (3,4), then X is uniform on
the set T of transcendentals in (3,4), because (3,4)\T has measure 0.

--r.e.s.

Relevant Pages

  • Re: comparing two randomizers
    ... Galathaea is talking about Kolmogorov complexity, aka algorithmic information content, aka compressibility, where a string is considered random if the shortest program that prints the string is longer than the string itself. ... But programs are known (pseudorandom generators) which produce strings that appear to be random, even though they're not; that is, there's no known general way of detecting that they're compressible unless someone reveals the algorithm+input to you. ... Of course this also casts doubt on the idea that there are any sources of true randomness: maybe radioactive decays would be easy to predict if we just knew the right formula. ... The digits of pi seem to be good pseudorandom numbers. ...
    (sci.math)
  • Re: Randomness from data
    ... >>thus having obtained 'randomness' justified? ... > sampled value using less bits only, even if the phenomena you're sampling is ... which one of the digits x one could justifiably collect ...
    (sci.crypt)
  • Re: Proof that Quantum Mechanics is not random!
    ... >> Johan van der Galien wrote: ... > equidistribution or uniform distribution as you call it. ... test for deviations from randomness. ...
    (sci.crypt)
  • Re: Randomness of digits within pi
    ... do not seem to occur as frequent as ... Each of them should only turn up once in one million digits on ... describes results from a number of tests on the randomness ... number and 6 not that lucky. ...
    (sci.math)
  • Re: PI random? Debate running in circles (you try making math jokes)
    ... So, you ask, "Are the digits of Pi randomly distributed ?" ... "Because the notion of randomness invokes the ... Your question about randomness of Pi is no different that the next one ...
    (sci.math)