Self-affinity of non-convex quadrangles

Determine whether every non-convex quadrangle is self-affine; whether each non-convex quadrangle is n-self-affine for every integer n above some quadrangle-dependent threshold; and whether such a threshold can be chosen uniformly for all non-convex quadrangles, or at least for all non-convex quadrangles, in analogy with the uniform threshold n0 = 5 for convex quadrangles.

Background

The paper proves that for every integer n ≥ 3 there exists an n-self-affine non-convex quadrangle, and that no non-convex quadrangle is 2-self-affine. These existence and nonexistence results do not determine which individual non-convex quadrangles are self-affine or how their possible self-affinity degrees behave as n increases. The authors explicitly leave several related questions open: universal self-affinity, eventual self-affinity for each fixed quadrangle, and the existence of a threshold independent of the quadrangle.

References

The previous results give a first access to the non-convex case, but leave many questions open. Is every non-convex quadrangle self-affine? Is there some non-convex quadrangle $Q$ and an integer $n_0=n_0(Q)$ such that $Q$ is $n$-self-affine for every integer $n \ge n_0$? Does the last apply to every non-convex $Q$? Does the last apply to every non-convex $Q$ with $n_0$ not depending on $Q$, as it does for convex $Q$ with $n_0=5$ \Theorem~2?

— Self-affine quadrangles  (2502.15521 - Richter et al., 21 Feb 2025) in Section Non-convex quadrangles, final paragraph