Superdensity and Super-Micro-Uniformity in Non-Integrable Flat Systems

The paper proves super-micro-uniformity for half-infinite geodesics on polysquare (square-tiled) surfaces with badly approximable slope by a careful density-transport/iteration argument leveraging a superdensity input, culminating in the bound |V_n(I) − n|I|/b| < ε n|I|/b for all subintervals |I| > C_ε/n, with C_ε independent of n. This is documented explicitly (statement of Theorem 1 and its proof, including the Case A/Case B dichotomy and the final estimate) . By contrast, the candidate solution asserts a much stronger O(1) absolute discrepancy via a cohomological-equation approach for Roth-type IETs applied to step functions. That would imply every interval is a bounded remainder set even for circle rotations (the b=1 case), which is known to be false in general (bounded remainder intervals are exceptional). The candidate’s use of MMY-type results is inapplicable to discontinuous step observables without resolving cohomological obstructions, and its reduction from “badly approximable slope” to the Roth/ bounded-type IET conditions is not justified.