Rotated odometers

The paper proves that for any permutation π on q symbols and any integer r ≥ q there is a finite-area translation surface L_{π,r} whose rational-slope q/r flow has Poincaré map measurably isomorphic to the rotated odometer F_π = a ∘ R_π, and whose metric completion has a single wild singularity plus at most finitely many cone-angle singularities; topologically, the surface has one non-planar end and possibly finitely many planar ends. This is established by first building a Chamanara-type surface L (horizontal identification by the von Neumann–Kakutani map a and vertical sides by translation), obtaining the Poincaré map F = a ∘ R_π for slope tan⁻¹(q/p) (Proposition 2.2), and then modifying vertical identifications to realize an arbitrary π and slope q/r (Proposition 2.3). The possibility of extra planar ends ("whiskers") is explicitly accounted for and tied to cone-angle singularities (via Randecker’s results), see Theorem 1.1 and the discussion around Proposition 2.3. In contrast, the model solution hard-wires F_π into the top–bottom gluing as G(x)=F_π(x−r/q) and claims that all horizontal endpoint-accumulation occurs only at the corners, yielding exactly one end. This is incorrect: shifting the countable discontinuity set of F_π by r/q generally creates accumulation at an interior point of the top edge, not only at the corners, and the model’s addition of a dyadic refinement D does not eliminate that interior accumulation. Moreover, the model simultaneously allows finitely many cone-angle singularities and yet asserts there is exactly one end, contradicting the cited correspondence between singularities and ends (each cone-angle singularity yields a planar end). The paper’s construction and claims are consistent and supported by detailed arguments and references; the model’s construction overlooks the accumulation geometry and ends–singularities correspondence.