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MathGroup Archive 2004

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An entropy measure of rational number fractal dimension: d=0.3732201657487591656832916543359

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50315] An entropy measure of rational number fractal dimension: d=0.3732201657487591656832916543359
  • From: Roger Bagula <tftn at earthlink.net>
  • Date: Thu, 26 Aug 2004 06:51:19 -0400 (EDT)
  • Reply-to: tftn at earthlink.net
  • Sender: owner-wri-mathgroup at wolfram.com

  I've done some other work using the Farey tree functions on the interval

[0,1].
In this case because of the singularity at Log[0], I limit
myself to single repetitions of fractions in the interval (0,1], zero
excluded.
As the number on points gets larger this average gets smaller but very 
slowly
so this is an upper bound for the dimension by an entropy measure of 
this type
( Ulam base two information entropy).
It is in the right range for the 3 set, 1/3 sets model:
1/3 rational numbers
1/3 irrational algebraic numbers
1/3 transcendental numbers
where rational numbers never touch transcendental numbers.
This program is a start of 49000 where duplicates are deleted and zero 
is deleted
to deplete that number in the end average.
It is an honest effort to make an estimate of the true fractal dimension 
of the rational numbers.


Mathematica notebook:

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Cell[BoxData[{
    \(f[a_, b_] := \((\((a/b)\)/\((1 - a/b)\))\) /; 0 <= a/b <= 1/2\), 
    \(f[a_, b_] := \((\((1 - a/b)\)/\((a/b)\))\) /; 1/2 < a/b <= 1\)}], 
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Cell[BoxData[
    \( (*\ symmetrical\ table\ of\ Farey\ rational\ numbers*) \)], "Input"],

Cell[BoxData[
    \(\(a = Table[If[n < m, f[n, m], f[m, n]], {n, 1, 700}, {m, 1, \ 700}]; 
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Respectfully, Roger L. Bagula

tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/ 





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