INDUCING COUNTABLE LEBESGUE SPECTRUM

The paper proves that every ergodic system induces a system with pure Lebesgue spectrum and infinite multiplicity (Theorem A) via a precise weak→strong regularization of spread-out densities (Propositions 4–5), an “import” lemma for multiplicity (Lemma 6), an inductive step (Proposition 8), and a multiplicity criterion (Lemma 2 and Proposition 3), culminating in Section 4’s density argument for purity. These ingredients and their interplay are clearly stated and coherent in the PDF (e.g., Theorem A and definitions in §2.1; Lemma 6; Proposition 8; Section 4) . By contrast, the candidate solution repeatedly misstates core definitions and the multiplicity criterion: it replaces the paper’s pα,τ-good/strong-closeness framework by ad hoc “η-good on an arc J” densities and relies on spectral projections to arcs J, attributing this to “AFT Lemma 2,” which the paper does not do—Lemma 2 works with fi and fi±fj and a direct-integral contradiction, not spectral band projections . The candidate’s claim that choosing an arbitrary arc J and then “using invariance” yields infinite multiplicity a.e. is not justified and departs from the paper’s actual Section 4 density scheme, which is what ensures pure Lebesgue type and combines with the multiplicity construction for the final result .