I'm looking for a particular class of connected, fractal sets $S_{\epsilon}$, with $0 < \epsilon < 1$ inside the unit disk.
The sets are defined such that the circle always belongs to the set $S$, the set has Hausdorff dimension $m$, and if one takes a random diameter $D$ splitting the disk in two, the following two things hold:
1) the Hausdorff dimension of $D \cap S$ is $n$
3) The exponents satisfy $m - n = \epsilon$
Question:
Do such fractals exist? What can be said about them?
You seem to require that the intersections with diameters and scaled disks are Lebesgue measurable (with respect to 2- or 1-dimensional measure, respectively). Let $f(\phi)$ denote the measure of the diameter in direction $\phi$ intersected with the fractal. For $r_1<r_2\le 1$ we find that for each diameter, the measure of intersection with the annulus with radii $r_1$ and $r_2$ is $(r_2^n-r_1^n)f(\phi)$. Then the area $A(r_1,r_2)$ within the annulus is bounded as follows: $$r_1\int_\phi (r_2^n-r_1^n)f(\phi)\,\mathrm d\phi \le A(r_1,r_2)\le r_2\int_\phi (r_2^n-r_1^n)f(\phi)\,\mathrm d\phi.$$ On the other hand, by the proportionality required for areas, we find $$ A(r_1,r_2)=A(0,r_2)-A(0,r_1)=(r_2^m-r_1^m)A(0,1).$$ We conclude $$ \frac{r_2^m-r_1^m}{r_1(r_2^n-r_1^n)}\ge\frac{\int_\phi f(\phi)\,\mathrm d\phi}{A(0,1)}\ge\frac{r_2^m-r_1^m}{r_2(r_2^n-r_1^n)}.$$ If $r_1\to r_2$, both bounds tend to $\frac mnr_2^{\epsilon-1} $, but the expression in the middle is a constant!