Notes on Symplectic Squeezing in T∗Tn and Spectra of Finsler Dynamics

The paper proves four statements (A–D) about symplectic embeddings in T* T^n and marked length spectra. For A–B it constructs, for each scale, explicit SL(n, Z) cotangent lifts built from Diophantine approximation (plus a mild rescaling), yielding embeddings P^{2n}(r) → Y^{2n}(1, v) and hence embeddings of any bounded domain, exactly as stated in Theorems A–B . For C it shows, contrary to the model’s assertion that a “one-shot linear argument fails,” that a single linear symplectomorphism (again in SL(3, Z)) does suffice when w admits a biased approximation; the proof controls two independent w^⊥-projections via an explicit matrix construction and a choice of k in (23)–(25) . For D the paper uses symplectic Banach–Mazur interleavings and barcode stability to deduce equality of the full marked length spectra from π~1(M)-trivial Liouville embeddings both ways, a mechanism absent from the model’s outline (which only sketches minimal-length monotonicity and invokes Floer identifications) . The model’s A–B are plausible via a different geometry-of-numbers route, but C contradicts the paper and leaves a key lattice estimate unproved; D is incomplete for the full marked spectrum because it lacks the barcode/interleaving step.