Squaring the circle via affine congruence by dissection with smooth pieces (Q2381803)

Let \({\mathcal D}^k\) denote the set of topological disks in \(\mathbb R^2\) with piecewise \(C^k\) boundaries \((k=1,2,\dots,\infty)\). A square \(S\) and a circular disc \(C\) are called congruent by dissection with respect to the group \(\mathbf{Aff}_+\) of orientation preserving affine transformations of \(\mathbb R^2\) (\(S \overset{k}{\sim} C\)) if there exist an integer \(n\geq 1\) and a dissection \(S=\sum_1^n S_i\) of \(S\) into \(n\) interior disjoint subsets \(S_i\in {\mathcal D}^k\), such that \(C=\sum_1^n \gamma_i(S_i)\) with some \(\gamma_i\in \mathbf{Aff}_+\). The author proves that \(S \overset{2}{\sim} C\) can be realized by only 6 pieces of dissection and \(S\overset{k}{\sim} C\) for all \(k\in \mathbb N\) with \(k\geq 3\) by 14 pieces. This improves an earlier result of the author and the reviewer [Beitr. Algebra Geom. 44, 47--55 (2003; Zbl 1040.52006)]. The proofs are quite tricky but elementary. The author conjectures that \(S \overset{\infty}{\sim} C\) is impossible.