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Topic: Cantor like set - with smaller "measure" ?
Replies: 3   Last Post: Feb 8, 2009 7:17 PM

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drobotv@gmail.com

Posts: 14
Registered: 5/21/07
Re: Cantor like set - with smaller "measure" ?
Posted: Feb 8, 2009 7:17 PM
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On Feb 7, 3:42Êam, bhawanvi...@gmail.com wrote:
> Hi,
>
> if X is the cantor set then, X + X = Ê{x+y:x,y belong to X} = [0,2]
>
> Is there an uncountable set, Y such that the measure of Y+Y is zero?
>
> Thanks,
> Bhawanvi


Well, actually more is true. Erdos and Volkmann construct a subgroup
of the real line (with addition as the operation) which has any
prescribed Hausdorff dimension, including dimension = 1, in which case
the subgroup is actually proper, of course. It fairly easily follows
than that such a subgroup must have measure 0. This is because a
result by Steinhaus which says that if E is a set of positive measure,
then E + E contains an open interval, so G would have to be the entire
line. The Math Reviews reference to the article is below:

MR0186782 (32 #4238)


As ever,


Vlad


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