dimension of a null set

From: Leonid V. Kovalev (lvkovale_at_artsci.wustl.edu)
Date: 11/19/04 

Date: 19 Nov 2004 07:45:03 -0500

The following is a textbook example of a set that has measure zero
but is of the 2nd Baire category.

Let {x_n : n=1,2..} be a dense subset of the interval [0,1].
Let {c_n : n=1,2..} be a convergent series with positive terms.
For each positive integer m let E_m be the union (over n)
of open intervals of length (c_n)/m centered at x_n.
Finally, let E be the intersection of all E_m. It is obvious
that E is a null set of the 2nd category.

But what about the Hausdorff dimension of E? Can it be positive (for
some values of x_n and c_n)? Can it be equal to 1? Any references
would be appreciated.

Sincerely,
Leonid Kovalev

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