Measure ratio of equidecomposable sets (Q1205425)

The paper deals with the problem of the paradoxical decomposition of balls (sets). The following is proved: For every \(n\), \(r\) such that \({2\over n}\leq r\leq{n\over 2}\) there are two sets \(A,B\subseteq R^{3}\) such that \({\lambda(B)\over\lambda(A)}=r\) and \(A\) and \(B\) are equidecomposable using \(n\)-pieces, i.e. there are partitions \(\{A_{1},\dots,A_{n}\}\) of \(A\) and \(\{B_{1},\dots,B_{n}\}\) of \(B\) such that \(A_{i}\) and \(B_{i}\) are isometric for each \(1\leq i\leq n\). Moreover a new proof of the ``inverse'' result of M. Laczkovich is given, namely that claiming that whenever \(A\) and \(B\) are equidecomposable using \(n\)-pieces then \({\lambda(B)\over\lambda(A)}\leq{n\over 2}\).