[FOM] Category and Measure

[joeshipman@aol.com]

***
----Original Message-----
From: James Hirschorn <James.Hirschorn at univie.ac.at>
 But perhaps the following example is relevant:

Let a_n and b_n be sequences of real numbers, indexed by the natural 
numbers.
Call them *similar* if one can be obtained from the other by 
translation and
dilation, i.e.

     a_n = r X b_n + s for all n, for some fixed reals r and s.

Consider the following statement: "Given a bounded sequence a_n of 
reals,
every Borel set of reals that is not 'small' contains a sequence 
similar to
a_n".

This statement with 'small' interpreted as meager is a theorem of 
Erdos. The
last I heard (several years ago) it is an open problem for 'small' = 
Lebesgue
measure zero.
***

Thanks, this is the kind of thing I was looking for, though it would be 
nice to see one which was actually settled as true for category-small 
but false for measure-small.

-- JS
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