Aperiodic and linearly repetitive Lorentz gases of finite horizon are not exponentially mixing

The paper proves that aperiodic, linearly repetitive (LR) Lorentz gases of finite horizon are not exponentially or stretched-exponentially mixing for any Hölder class and derives the sharp restriction γ ≤ 2(1+α) on any polynomial decay, by constructing “far away” characteristic observables supported on well-separated LR tower boxes and using finite-horizon geometry to force long time intervals with non-vanishing correlations; see the statement of Theorem 1 and the setup of the compactified collision space D, the tower-system structure, and the key bounds (5) and (6) leading to the contradiction with fast mixing and the polynomial restriction . By contrast, the model’s solution hinges on two unsupported steps: (i) it asserts a base “return” π ∘ F^{n_R} = π for suitably chosen n_R from linear repetitivity, which would amount to a translation-periodicity of patches and is not implied by LR in an aperiodic setting; and (ii) it posits a uniform, R-independent crossing/measure lower bound for h_R := I ∘ F^{n_R} on thin boxes, which overlooks the growing complexity and potential distortion near billiard singularities for large n_R. These issues invalidate the model’s proof, even though its final numerical restriction coincides with the paper’s theorem. The paper’s argument is internally consistent and grounded in the LR tower framework and finite-horizon geometry, and does not rely on the problematic steps in the model’s approach .