TITLE: Paradoxical spherical caps
QUESTION [8 upvotes]: All spherical caps (i.e. sets $C_L:=\{(x,y,z)|x^2+y^2+z^2=1, z\geq L\}$) admit a paradoxical decomposition in the sense of Banach-Tarski, meaning $C_L \tilde{} 2C_L$; here $\tilde{}$ stands for the relation of equivalence by finite equidecomposition.  (See http://www.maths.usyd.edu.au/u/athomas/amenability/Lecture5_Spheres.pdf)
Since flat disks do not admit paradoxical decompositions, a compactness argument suggests that no universal upper bound exists for the number of pieces necessary for a paradoxical decomposition of a spherical cap.
So what upper and lower bounds are known for the minimum necessary number of pieces, as a function of $L$?

REPLY [4 votes]: There are absolute lower and upper (we assume here that we do not make the pieces smaller by crushing) bounds on the number of pieces in the paradoxical decompositions in question. Namely:
if $L=-1$, the sphere is paradoxical using 4 pieces;
if  $L \in (-1,0\rangle$, the cap $C_L$ is paradoxical using 5 pieces;
if $L \in (0,1)$, the cap $C_L$ is paradoxical using 6 pieces.
This follows from the papers of R. M. Robinson "On the decomposition of spheres", Fund. Math. 34 (1947), 246-260 and of G. A. Sherman "Minimal paradoxical decomposition for Mycielski's square", Fund. Math. 139 (1991), 151-165, who proved also that the paradoxical sets contained in the caps $C_L$ use the same number of the pieces in the decompositions as $C_L$, where $L$ runs through $\langle-1, 1\rangle $.