Staircase behavior for scalar \(Z_n\)-equivariant embeddings

Determine whether, for every positive integer \(n\), the scalar \(Z_n\)-equivariant ellipsoid embedding problem in \(\mathbb C^2\) exhibits staircase behavior governed by the recursion \(l_{i+2}=(n+2)l_{i+1}-l_i\).

Background

The scalar action of ZnZ_n on C2\mathbb C^2 is given by μ⋅(z1,z2)=(μz1,μz2)\mu\cdot(z_1,z_2)=(\mu z_1,\mu z_2). The paper places its staircase results for the n=2n=2 and n=4n=4 cases within the family of scalar cyclic-group actions, while the n=1n=1 case is the classical Fibonacci staircase.

The unresolved question is whether all scalar cyclic-group actions produce analogous infinite staircases and whether the staircase sequences are universally determined by the stated second-order recursion.

References

Is this so for all $n \in N_{>0}$, i.e.\ do all $Z_n$-equivariant embedding problems exhibit staircase behaviour and are they governed by the recursion $l_{i+2}=(n+2)l_{i+1}-l_i$?

— Quantitative symplectic topology of Katok's examples and equivariant symplectic embeddings  (2609.35209 - Adaloglou et al., 28 Sep 2026) in Section 1, subsection “Open questions,” item (b)