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




Re: Cantor like set  with smaller "measure" ?
Posted:
Feb 8, 2009 7:17 PM


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
=================== * Vladimir Drobot * Retired and gainfully unemployed * http://www.vdrobot.com * mailto:vlad@vdrobot.com ==================



