From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Measure 0 subfield of R
Date: 4 Sep 2000 22:10:37 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8p0lgs$8il$1@nnrp1.deja.com>, FGD  <presbeis@my-deja.com> wrote:
>The following question recently appeared on the French group
>fr.sci.maths (author Pierre Bernard),

>   Is there a measure 0 subfield of R with power |R|?


Let S be an uncountable closed set of Hausdorff dimension 0 (these 
can be constructed by a modification of the Cantor middle-thirds 
construction), and K the field it generates.
  
K is the union of sets K_n where K_0 = S and 
K_{n+1} = {a, a + b, a - b, a b, a/b (if b <> 0): a, b in K_n}.
It's not hard to show that if K_n has Hausdorff dimension 0, then
so does K_{n+1}.

So K is a field of power |R| and not only has measure 0 but also 
Hausdorff dimension 0.

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