  Algorithmic fractal dimensions quantify the algorithmic information density
of individual points and may be defined in terms of Kolmogorov complexity. This
work uses these dimensions to bound the classical Hausdorff and packing
dimensions of intersections and Cartesian products of fractals in Euclidean
spaces. This approach shows that two prominent, fundamental results about the
dimension of Borel or analytic sets also hold for arbitrary sets.
