DENSITY OF GROUP LANGUAGES IN SHIFT SPACES

The paper’s Theorem 3.3 gives δ_μ(L) = ∑_{k∈K} ∑_{g∈G} μ({g}×X) μ({gk}×X) for any ergodic lift to the skew product, and proves it by expressing μ(L∩A^i) as a finite sum of correlations and applying the mean ergodic theorem in Cesàro form. The candidate solution uses the Koopman-operator formulation of the same Cesàro correlation limit. Aside from minor presentational differences (operator inner products vs. set intersections; reduction to singleton K in the paper), the arguments are the same at core and yield the identical formula, including independence of the ergodic lift. The only omission in the candidate solution is not explicitly justifying existence of an ergodic lift (handled in the paper by Lemma 3.4).