Orientation preserving maps of the square grid II (Q6624168)

The paper investigates orientation-preserving mappings of the square grid \(G_n = \{0, \pm 1, \ldots, \pm n \}^2\) in the Euclidean plane. A map \(\varphi: S \rightarrow \mathbb{R}^2\), where \(S \subset \mathbb{R}^2\) is a finite set, is defined as orientation preserving if, for every non-collinear triple \(u, v, w \in S\), the orientation of the triangle formed by \(u, v, w\) is the same as that of the triangle formed by \(\varphi(u), \varphi(v), \varphi(w)\).\N\NThe authors demonstrate that for sufficiently large \(n\), any orientation-preserving map \(\varphi : G_n \rightarrow \mathbb{R}^2\) can be closely approximated by a projective transformation \(\mu : \mathbb{R}^2 \rightarrow \mathbb{R}^2\). Specifically, they show that there exists such a \(\mu\) satisfying \(\| \mu \circ \varphi(z) \| = \mathcal{O}(1/n)\) for every \(z \in G_n\). The bound \(\mathcal{O}(1/n)\) is the best possible, which is shown by an example presented in the paper.\N\NMoreover, the authors prove that for every \(k \in \mathbb{N}\) and for every \(\varepsilon \in (0, 0.1)\) there is \(n\) such that if \(\varphi: G_n \rightarrow \mathbb{R}^2\) is an orientation preserving map, then there is an affine transformation \(\alpha: \mathbb{R} \rightarrow \mathbb{R}^2\) and \(a \in \mathbb{Z}^2\) such that \(a + G_k \subset G_n\) and for every \(z \in a + G_k\), we have \(\| \alpha \circ \varphi(z) - z\| < \varepsilon\).