^{n}, I had an idea.

I'll recall the "non-formal" definition of Hausdorff-Besicovitch Dimension (that doesn't work in all cases) :

Take a set A, and apply on it an homothetic transformation with a scale k (u(A)). We define n as the quotient between the measure of u(A) and A.

And we define the number ln(n)/ln(k) as the Hausdorff-Besicovitch Dimension of A.

Example 1 : A is a point, and whatever the homotetic transformation we apply to it, it gives a point, that contains only one times itself, so the dimension of a point is ln(1)/ln(whatever)=0.

Example 2 : A is a line. If we multiply the lengths by a factor 2, we get a new lint that can contain two times itself, its dimension is ln(2)/ln(2)=1.

It works also with a square (ln(4)/ln(2)=2), with a cube, with the Kock snowflake, and with the Menger sponge.

So I tried to apply it on a discrete group of ℝ (a group of number of the form xn, where x is a fixed number of ℝ* and n∈ℤ) and we take a definition of the measure of the discrete sets. With a transformation of factor 2, it gives the sets of the 2xn. And the original set can contain two times the new set, so its dimension should be : ln(1/2)/ln(2)=-1 !

So I thought that negative dimensions could be the dimensions of countable infinite sets.

Another example (very strange) : consider a set made of the union of an infinity of parallel lines (that are equidistant), like this :

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Double its size :

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We can cut now the red part, and paste it like this :

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So, doubling its size doens't change its measure, so its dimension is ln(1)=ln(2)=0.

What do you think about this ? Could a solid definition of negative dimensions can be done ?