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.

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