Simple paradoxical replications of sets (Q1592513)

Equidecomposition of \(A\) and \(kA\), the latter standing for \(k\) replicas of \(A\), is given if there are partitions \((A_1,\dots,A_n)\), \((B^1_1,\dots,B^1_{r_1})\),\dots, \((B^k_1,\dots,B^k_{r_k})\) of \(A\) such that \(A_{r_1+\dots+r_l+s}\) can be mapped isometrically onto \(B^{l+1}_s\). The smallest \(n\), for which such decompositions exist, is denoted by \(\text{deg}(A,kA)\). From Theorems 4.5 and 4.7 of \textit{S. Wagon} [``The Banach-Tarski paradox'' (1986; Zbl 0569.43001)] the introduced quantity \(\text{deg}(A,kA)\) is computed for \(A\) standing for the spheres or balls in \(\mathbb R^3\). Further, the upper estimate \(\text{deg}(A,kA)\leq 32k+1\) is obtained for bounded convex bodies in \(\mathbb R^3\). For convex bodies with additional properties better estimates are derived. In particular, \(\text{deg}(A,kA)\leq 8k-3\) if \(A\) is a cube in \(\mathbb R^3\).