Khintchine dichotomy for self-similar measures

The paper proves Theorem A (the full Khintchine dichotomy for all self‑similar measures on R) by establishing an effective equidistribution statement (Theorem B, with polynomial rate t^{-c}) and then deriving both the convergence and divergence cases via a refined adaptation of the Khalil–Luethi mechanism in Section 6, including new correlation estimates (e.g., Proposition 6.6) and short-range quasi-independence (Proposition 6.7) . The candidate solution follows the same overall strategy (Dani correspondence → equidistribution along horocycles → Khalil–Luethi implication) and correctly identifies that Theorem B feeds into Theorem A via Khalil–Luethi, noting that Section 6 handles the needed adaptations beyond contractive/open-set assumptions . However, it misstates the rate in Theorem B as exponential e^{-κt} (the paper proves a polynomial rate t^{-c}) and informally invokes “summable in t” error terms; the paper instead obtains the necessary estimates via correlation decay and quasi-independence. After correcting these technical points, the model’s proof aligns with the paper’s approach and conclusions.