HÖLDER CONTINUITY OF THE INTEGRATED DENSITY OF STATES FOR LIOUVILLE FREQUENCIES

The paper proves Hölder continuity of the IDS at energies satisfying L(ω,E) > 4 κ(ω,E) β(ω) via a new effective LDT built from a Riesz–mass/annulus Green’s function analysis and Avila’s acceleration, then uses a resolvent block argument to convert this LDT into a Hölder bound. The candidate solution reaches the same conclusion using a different outline: it posits an LDT along complex phase lines and a three-lines transfer back to the real axis, plus the same resolvent-block scheme. While the model misattributes part of Han–Schlag’s method (their LDT is on the real phase and obtained by Fejér smoothing and Fourier/Riesz control, not by a two-lines lemma), the claimed implication and final result match the paper. Hence both are correct, but the approaches are materially different.