From: "G. A. Edgar" <edgar@math.ohio-state.edu.nospam>
Subject: Re: A countable set with fractal dimension D > 0
Date: Tue, 16 May 2000 13:15:25 -0400
Newsgroups: sci.math
Summary: [missing]

The Hausdorff dimension of a countable set is zero.  (So is the
packing dimension.  And many others.)  The Bouligand dimension
(box dimension) of a set is the same as for the closure of
the set.  So, for example, the Bouligand dimension of
the set of rationals is 1.  And the countable set of endpoints
of the Cantor set has Bouligand dimension equal to log 2/log 3.
When you say "fractal dimension" you probably should say
what you mean.

For "similarity dimension" you will have to say what you mean
by it.  In my book
 <http://www.math.ohio-state.edu/~edgar/books/mtfg.html>
the term "similarity dimension" really applies to an iterated
function system rather than to the invariant set of
the IFS.  In order for the similarity dimension to be
useful in the study of a set, the set should be the attractor
of the IFS, so in particular it should be a nonempty compact
set.  Otherwise (as in your case of the rationals Q, or even
the real line R) you will end up with useless answers like:
R has similarity dimension 0.

Here's another "similarity dimension" computation that I saw once:
Consider the set Z of integers.  It is the disjoint union of
the even integers and the odd integers.  Each of these sets
is similar to Z itself with ratio 2.  So to find the similarity
dimenssion, we solve 2*2^s = 1 to get dimension s = -1.

-- 
Gerald A. Edgar              edgar@math.ohio-state.edu
==============================================================================

From: Jan-Christoph Puchta <jcp@arcade.mathematik.uni-freiburg.de>
Subject: Re: A countable set with fractal dimension D > 0
Date: Tue, 16 May 2000 19:22:08 +0200
Newsgroups: sci.math
To: David Sanchez Molina <david.sanchez-molina@upc.es>
Summary: [missing]

David Sanchez Molina wrote:
> 
> A countable set with fractal dimension D > 0.
> 
> Paradoxically the set constructed in this message
> is countable and it has similarity [fractal]
> dimension D > 0, but I heard fractal dimension of
> a countable set is always zero. I present here the
> 
> construction for your judgement:
> 
> The set in question consists of all points of the
> form (in base 3):
> 
> 0.a1-a2-...-an  [where each ai is precisely 0 or
> 2, and n finite]

The Hausdorff-dimension of a set equals the dimension of
its closure, e.g. the rational numbers are obviously self
similar with dimension 1. Hence the statement quoted above
might only talk about closed sets. Note that the closure
of your set is exactly the Cantor set.


JCP
==============================================================================

From: israel@math.ubc.ca (Robert Israel)
Subject: Re: A countable set with fractal dimension D > 0
Date: 17 May 2000 07:28:44 GMT
Newsgroups: sci.math
Summary: [missing]

In article <392183C0.167E@arcade.mathematik.uni-freiburg.de>,
Jan-Christoph Puchta  <jcp@arcade.mathematik.uni-freiburg.de> wrote:

>The Hausdorff-dimension of a set equals the dimension of
>its closure, e.g. the rational numbers are obviously self
>similar with dimension 1.

No, and no.  The Hausdorff dimension of a countable set is
always 0. Take a look at the definitions of Hausdorff
measure and dimension (which BTW do NOT involve self-similarity),
and this is obvious: for any r > 0 and delta > 0, the size
delta approximating measure of your set is 0, because you can
cover your set by a countable family of closed sets of diameter
less than delta such that the sum of the r'th powers of the
diameters of these sets is 0 (namely the closed sets can be single
points).

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2