Sarnak’s Conjecture for Rank-One Subshifts

Part (a) of the candidate solution follows the paper’s argument for accc rank‑one subshifts, using Stone–Weierstrass and the KLR short‑interval estimate to handle intersections of translates of Mx; this is essentially Theorem 2.2 and Theorem 2.4 in the paper, culminating in Theorem 1.9 that accc rank‑one subshifts are Möbius disjoint . However, in part (b) the candidate asserts that for the generalized Katok class K every Mx is an accc “with k = m.” The paper does not claim this and, in fact, proves Möbius orthogonality for K by approximating with unions of arithmetic progressions whose common differences are q ≈ |vN1| + p (at most m distinct moduli), not by a single fixed modulus m; see Theorem 3.1 and its proof, and Corollary 3.2 . Thus the candidate’s reduction “K ⇒ accc with k = m ⇒ (a)” is unsupported. The paper’s proofs (via KLR, Theorem 1.10) are correct as written .