|City University of Hong Kong
Any set containing a sphere centered at every point cannot have 0 Lebesgue measure. This is a consequence of the L^p boundedness of the spherical maximal function. On the other hand, there exist sets of 0 Lebesgue measure which contain a large family of spheres, which may be considered as Kakeya/Nikodym sets for spheres. We will discuss such sets and their Hausdorff dimension, and related estimates for maximal functions associated with spheres.