  In his celebrated paper on area distortion for quasiconformal mappings,
Astala showed optimal area distortion bounds and dimension distortion estimates
for planar quasiconformal mappings. He asked (Question 4.4) whether a finer
result held, namely absolute continuity of Hausdorff measures under
push-forward by quasiconformal mappings. This was proved in one particular case
relevant for removability questions, in joint work of Astala, Clop, Mateu,
Orobitg and the author ("Distortion of Hausdorff measures and improved
Painlev'e removability for bounded quasiregular mappings", Duke Math J., to
appear [ACMOU]) (Theorem 1.1), the other cases remaining open. A related
question that we left open in [ACMOU] (Question 4.2) (which was asked by Astala
to the author before [ACMOU] in an equivalent form in a personal communication)
is whether BMO removability for K-quasiregular mappings and ($L^{\infty}$)
removability for K-quasiregular mappings are indeed different problems.
  In this paper we give a series of examples answering in the positive Question
4.2 in [ACMOU], at the same time proving sharpness in two different senses of
Theorem 1.1 in [ACMOU], and also giving examples that would yield sharpness in
those two different senses as well for the absolute continuity of Hausdorff
measures under push-forward by quasiconformal mappings, were it to be proved.
