On a Problem in Geometric Measure Theory Related to Sphere and Circle Packing

In this thesis we prove that a Borel set which contains spheres centered at all points of a Borel set of Hausdorff dimension greater than 1 must have positive Lebesgue measure, and, using the same method, we rederive a special case of Stein's spherical means maximal inequality. We also prove the corresponding result for circles, provided that the set of centers has Hausdorff dimension greater than 3/2.