From: "r.e.s."
Subject: Re: continued fraction terms for pi
Date: Wed, 2 Jun 1999 11:12:46 -0700
Newsgroups: sci.math
Keywords: Gauss-Kusmin distribution of partial quotients
Thank you for the Mathematica output.
Here's a summary of what it shows for
integers 1,...,10 among the first 10,000 terms:
---------------------------------------------
obs. obs. Gauss-Kusmin
k freq. proportion probability discrepancy
---------------------------------------------
1 4206 0.4206 0.415037499 0.005562501
2 1672 0.1672 0.169925001 -0.002725001
3 883 0.0883 0.093109404 -0.004809404
4 597 0.0597 0.058893689 0.000806311
5 442 0.0442 0.040641984 0.003558016
6 282 0.0282 0.029747343 -0.001547343
7 224 0.0224 0.022720077 -0.000320077
8 186 0.0186 0.017921908 0.000678092
9 143 0.0143 0.014499570 -0.000199570
10 123 0.0123 0.011972642 0.000327358
---------------------------------------------
The rightmost two columns show the "fit" of
probability = log(1+1/(k*(k+2)) / log(2)
which is the limiting distribution
lim pr(a_n=k) (n->oo)
appearing in the Gauss-Kusmin theorem.
See
http://www.mathsoft.com/asolve/constant/kuzmin/kuzmin.html
(This may be surprising, since the applicability
of that theorem my be questioned in the present
context.)
--
r.e.s. (Spam-block=XX)
Jim Ferry wrote ...
: Also of possible interest: the frequencies of the terms
: in the first 10, 100, 1000 and 10000 terms of the continued
: fraction for pi (= [3,7,15,1,292,1,1,1,2,1,...]). I'd
: expect the term n about 1/(n(n+1)) of the time (based on
: an obvious, possibly naive heuristic) for a generic number.
: Based on this, pi seems a little light on 1's.
: In[7]:= a[10000]
: Out[7]= {{4206, 1}, {1672, 2}, {883, 3}, {597, 4}, {442, 5},
: {282, 6}, {224, 7}, {186, 8}, {143, 9}, {123, 10}, {78, 11},
==============================================================================
From: "r.e.s."
Subject: Distribution of Continued Fraction Terms for Pi
Date: Wed, 2 Jun 1999 16:51:04 -0700
Newsgroups: sci.math
Here's something remarkable about the empirical distribution
of continued fraction terms for pi: The Gauss-Kumin limit-
distribution fits very well -- but why should it?
I believe that what we have here is the continued fraction
parallel to the well-known pseudo-randomness in the decimal
digits of pi.
The relative frequencies of the first 10,000 terms for pi,
according to Mathematica output posted by Jim Ferry in
another thread, provide the following:
------------------------------------------------
obs. obs. Gauss-Kusmin
k freq. proportion probability* discrepancy
------------------------------------------------
1 4206 0.4206 0.4150 0.0056
2 1672 0.1672 0.1699 -0.0027
3 883 0.0883 0.0931 -0.0048
4 597 0.0597 0.0589 0.0008
5 442 0.0442 0.0406 0.0036
6 282 0.0282 0.0297 -0.0015
7 224 0.0224 0.0227 -0.0003
8 186 0.0186 0.0179 0.0007
9 143 0.0143 0.0145 -0.0002
10 123 0.0123 0.0120 0.0003
>10 1242 0.1242 0.1255 -0.0013
------------------------------------------------
* probability = log(1+1/(k*(k+2)))/log(2)
------------------------------------------------
A simple goodness-of-fit test suggests that
discrepancies of this magnitude or greater
would be expected in at least 60% of random
samples if the Gauss-Kumin limit-distribution
were in fact the underlying distribution here.
Theorem (Gauss-Kusmin):
If x is uniformly distributed on the interval (0,1)
and a_n(x) is the n-th continued fraction term for x,
then lim(n->oo)(pr(a_n(x) >= k) = log(1+1/k)/log(2)
for k=1,2,3,...
(The limit distribution can also be written using
pr(a_n=k) = pr(a_n>=k) - pr(a_n>=k+1)
-> log(1+1/(k*(k+2)))/log(2).)
But here's the peculiarity:
To realize pr(a_n(x)=k) "empirically", one would
generate x_1,x_2,... iid x, and examine the long-run
proportion of these x_i for which a_n(x_i)=k, for
sufficiently large n.
This is in contrast to what we have in the observed
distribution of terms for one particular x, namely
for x = fractional part of pi.
In a nutshell:
It is not clear why the empirical distribution of
a_1(x), a_2(x),... --i.e.*one x* & *multiple n*--
should so well approximate what is expected of the
empirical distribution of
a_n(x_1), a_n(x_2),... --i.e.*one n* & *multiple x*.
In other words, it is not clear why the successive
terms for pi mimic a sequence formed by taking a
single "remote" term from each of a succession of
numbers drawn uniformly at random from (0,1).
("Remote" meaning "sufficiently far into the sequence
of terms for that number".)
But the consequence seems to be that pi has a
continued fraction behavior that parallels the
well-known pseudo-randomness in its decimal digits;
if so, it would seem to have a more elegant flavor,
since the continued fraction terms of pi are unique
and independent of numerical representation.
---
r.e.s. (Spam-block=XX)
==============================================================================
From: gerry@mpce.mq.edu.au (Gerry Myerson)
Subject: Re: Distribution of Continued Fraction Terms for Pi
Date: Thu, 03 Jun 1999 15:51:15 +1100
Newsgroups: sci.math
In article <7j4g18$j28@sjx-ixn9.ix.netcom.com>, "r.e.s."
wrote:
=> Here's something remarkable about the empirical distribution
=> of continued fraction terms for pi: The Gauss-Kumin limit-
=> distribution fits very well -- but why should it?
=>
=> Theorem (Gauss-Kusmin):
=> If x is uniformly distributed on the interval (0,1)
=> and a_n(x) is the n-th continued fraction term for x,
=> then lim(n->oo)(pr(a_n(x) >= k) = log(1+1/k)/log(2)
=> for k=1,2,3,...
=>
=> In other words, it is not clear why the successive
=> terms for pi mimic a sequence formed by taking a
=> single "remote" term from each of a succession of
=> numbers drawn uniformly at random from (0,1).
=> ("Remote" meaning "sufficiently far into the sequence
=> of terms for that number".)
Gauss-Kuzmin can also be twisted to say that for all reals x outside of
a set of measure zero the frequency of k as a partial quotient in the
expansion of x is log(1+1/(k*(k+2)))/log(2). That set of measure zero
contains all the rationals & the quadratic irrationals, but there's no
reason to think it contains pi (and empirical evidence suggests that
it isn't).
Gerry Myerson (gerry@mpce.mq.edu.au)
==============================================================================
From: "r.e.s."
Subject: Re: Distribution of Continued Fraction Terms for Pi
Date: Thu, 3 Jun 1999 02:13:35 -0700
Newsgroups: sci.math
Gerry Myerson wrote ...
[most of previous article quoted --djr]
This seems more than a merely semantic "twisting".
All sources I've found for the Gauss-Kuzmin theorem
refer to a *limit* distribution equivalent to
lim(n->oo) pr{x: a_n(x)=k} = log(1+1/(k*(k+2)))/log(2),
k=1,2,..., where a_n(x) = n-th term (partial quotient)
of the continued fraction for x.
How can this be seen as addressing the distribution of
terms in the continued fraction of a *particular* x?
(It's a *set* of x that has the limit-probability.)
I vaguely recall this as a kind of ergodic property.(?)
--
r.e.s. (Spam-block=XX)
==============================================================================
From: "r.e.s."
Subject: Re: Distribution of Continued Fraction Terms for Pi
Date: Thu, 3 Jun 1999 12:03:37 -0700
Newsgroups: sci.math
Keywords: Gauss-Kuzmin theorem
OK, I've convinced myself that the theorem you state
is at least made plausible by Gauss-Kuzmin in the
form I cited it.
For example, using "term" to mean "partial quotient",
an instance of the Gauss-Kuzmin theorem is that
there exists an N such that for any n>N,
"1.00...%" of all x in (0,1) have their n-th term
equal to eleven. (log_2(1+1/(11*13))=0.0100...)
But since this is true "for any n>N", we can say
loosely that 1% of *all* the "remote" terms of
(almost) *all* x in (0,1) are equal to eleven.
And the "almost" is reasonable because the original
statement about "1.00...%" could be true even if
there were an exceptional set of Lebesque measure 0.
(But I do think you need to state the result as
a limit for n->oo, i.e. to refer to sufficiently
"remote" terms in the partial fractions.)
And I wonder what it would take to make the argument
rigorous. (I still suspect ergodicity is involved.)
--
r.e.s. (Spam-block=XX)
r.e.s. wrote ...
[previous article --djr]
==============================================================================
From: "r.e.s."
Subject: Re: 355/113 Is there a better...
Date: Fri, 16 Jul 1999 17:41:05 -0700
Newsgroups: sci.math
Kurt Foster wrote ...
[snip]
: A great many partial quotients in the SCF for pi are known (but not by
: me). Perhaps one of the later ones is 292 or greater; this might give a
: larger value of c in (2) for pi. Anyone have that info handy?
In the first 2,000,000 terms (partial quotients),
about 10,000 (~0.49%) are >=292.
For the first 2,000,000 terms, I've confirmed that
the proportion of terms >=k is log(1+1/k)/log(2),
k=1,2,..., to within bounds that would be expected
if this were indeed the underlying distribution.
This is the Gauss-Kusmin distribution, which would
be the limit distribution for the terms if pi were
in fact normal.
--
r.e.s.
XXrs.1@mindspring.com (Spam-block=XX)