  Quasiconformal maps in the plane are orientation preserving homeomorphisms
that satisfy certain distortion inequalities; infinitesimally, they map circles
to ellipses of bounded eccentricity. Such maps have many useful geometric
distortion properties, and yield a flexible and powerful generalization of
conformal mappings. In this work, we study the singularities of these maps, in
particular the sizes of the sets where a quasiconformal map can exhibit given
stretching and rotation behavior. We improve results by
Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and
rotation sets with non-sigma-finite measure at the appropriate Hausdorff
dimension. We also improve this to give examples with positive Riesz capacity
at the critical homogeneity, as well as positivity for a broad class of gauged
Hausdorff measures at that dimension.
