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I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.

For simplicity, I was hoping to give a more intuitive introduction using 'squared paper': suppose you draw your figure on squared paper with squares of size $2^{-n}$, and let $M(n)$ the number of squares that your figure touches; take the limit of $M(n)(2^{-n})^d$ for $n\to\infty$.

This seems to be ok for defining the dimension $d$ (the only issue is that you only allow coverings with very specific sets, the 'elementary squares'), but not for the measure. In fact, if you just consider line segments, then this definition converges to their 1-norm length ($|x_1-x_0|+|y_1-y_0|$) instead of the usual Euclidean length.

So, my questions are:

1- is there any hope to save this approach, or do I really have to introduce the measure using balls (this would be unpleasant as I would also have to deal with the proper constants for the volume of a ball in fractional dimension, ouch)

2- is there a simpler introduction that I am overlooking here?

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2 Answers

up vote 4 down vote accepted

The discussion at http://en.wikipedia.org/wiki/Box-counting_dimension (Minkowski–Bouligand dimension) seems fairly thorough.

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As Charles said, this would be the box-counting dimension (a.k.a. Bouligand dimension or Minkowski dimension). Not the Hausdorff dimension. And for the measure it would usually not converge so that is out.

For the Hausdorff dimension you don't have to use balls, you can use any sets, but you DO have to cover by sets of differing sizes. If you use covers by sets of the same size, you get the box-counting dimension.

In many cases the two dimensions are the same, but not in all cases.

For the Hausdorff dimension it is enough to compute it up to a constant, so you do not need to introduce a constant for the volume of a ball. And nowadays for the Hausdorff measure, we usually normalize using constant 1 in the definition.

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I can only accept one answer, but this is really helpful, too. Thanks to both you and Charles Matthews, this shows very clearly what the problem is. –  Federico Poloni Aug 12 '10 at 12:18
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