Is the measure induced by the Mandelbrot set computable on rational rectangles? Is there a computable function that, given a positive rational number $\epsilon$ and a rectangle with rational corners $A$ returns a number $f(A,\epsilon)$ such that $|\mu(A \cap M)-f(A,\epsilon)|\lt\epsilon$, where $M$ is the Mandelbrot set and $\mu$ is Lebesgue measure?
 A: It is (as far as I know) an unsolved problem.
It is equivalent to asking if the area of the Mandelbrot set is computable.  The connection is a follows:  By doing some careful interval arithmetic based on the usual construction of the Mandelbrot set, we can enumerate an open cover of the exterior.  After enumerating enough pixels, we would be able to account for all but $\epsilon$ of the area of the exterior, if we knew the area of the Mandelbrot set.  Then we would be able to approximate any pixel's intersection with the Mandelbrot set with within $\epsilon$.
Now, while it is possible to directly calculate the Mandelbrot set area (we have a series formula, but not a rate of convergence for that series), it is also likely that this problem will be solved instead by solving two unknown problems of the Mandelbrot set:

*

*Are the hyperbolic components the only interior regions?  If so, this would allow us to similarly enumerate the interior, since we know how to enumerate the hyperbolic components.

*Is the area of the boundary of the Mandelbrot set zero?  If so, then the interior and exterior would sum to full measure.  Therefore, to compute the area of the Mandelbrot set (or of the set inside any pixel) within $\epsilon$ it is enough to enumerate all but $\epsilon$ of the interior and exterior area.

