{"id":"be672a27-f4de-431a-83e5-dcef9b4139d2","arxiv_id":"1908.07554","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"Every zero-entropy sequence is Cesàro-close to a zero-entropy Toeplitz sequence, so Sarnak's conjecture reduces to the Toeplitz case.","lead":"The authors prove that any sequence over a finite alphabet can be approximated, in Cesàro average, by a Toeplitz sequence whose entropy is at most twice the original. This lets them reduce the Sarnak conjecture on Möbius disjointness to the special case of zero-entropy Toeplitz systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 states lim but the proof only gives limsup; the symmetric Cesàro average of |a-b| need not converge, so the theorem should be restated with limsup.","rationale":"The main mathematical core of the paper is the construction of a Toeplitz sequence approximating an arbitrary finite-alphabet sequence with entropy at most doubled. I read that construction and the entropy estimate carefully; the inequalities are plausible and the factor 2 comes from a square in the block count. The use of the Boyle-Downarowicz theorem in Section 1.5 is legitimate: it is a known theorem and the reduction to subshifts is standard. The finite-alphabet reduction for arbitrary continuous functions is not spelled out but is folklore and does not threaten the argument. The genuine soft spot is the final limit statement. The proof bounds each step's average difference and sums them, so it controls the limsup of the total average difference. It does not establish convergence of the Cesàro averages, and there is no reason the limit must exist for an arbitrary a; the first step's contribution can oscillate. The theorem as stated with lim is therefore not proved. The Sarnak application only needs the limsup, because if the limsup of the correlation difference is bounded by an arbitrary ε, the correlation tends to 0. Hence the central reduction survives a simple restatement, and the reader's CONDITIONAL verdict is appropriate. I mark agreement as partial because the reader's stated weakest assumption (Boyle-Downarowicz) is not the real concern; the limsup issue appears in the reader's rationale but not as the weakest assumption.","tokens_in":5700,"tokens_out":37926,"duration_ms":843797,"concrete_test":"Take k=2, fix ε, choose l1 large, and define a on multiples of l1 as a_{l1 m} = a_0 + 1 for m in blocks of length 2^{2j+1} and a_{l1 m} = a_0 for blocks of length 2^{2j+2} (symmetric on negative m). Run the paper's construction with this a and compute D_N = (2N+1)^{-1} Σ_{|n|≤N} |a_n-b_n| for N in geometric progression. If D_N oscillates (liminf < limsup), the limit in Theorem 1.1 fails for the constructed b and the proof only supports limsup. More directly, re-derive the theorem's final inequality without taking a limit; it yields limsup ≤ Σ ε_j < ε, not lim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 claims the existence of a Toeplitz b with lim_{N→∞} (2N+1)^{-1} Σ_{|n|≤N} |a_n-b_n| < ε. The proof bounds each step by ε_j and sums to obtain that the average is < ε for every N (or at least eventually). This establishes at most limsup_{N→∞} ... ≤ ε; it does not prove the limit exists. For a general sequence a, the constructed b's difference sequence can have non-convergent Cesàro means: the first overwriting step sets positions on l1Z to a_0, so the average difference includes (1/(2N+1)) Σ_{m: |l1 m|≤N} |a_{l1 m}-a_0|, which need not converge for arbitrary a (e.g., choose a on l1Z with alternating long blocks). Thus the stated theorem is not proven; only the limsup version follows. The Sarnak reduction in Corollary 1.2 requires only the limsup bound, because the Möbius correlation difference is bounded by the average absolute difference, and an arbitrary ε>0 forces the limsup to 0. So the main application survives a restatement.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for every finite-alphabet bi-infinite sequence a and every ε>0, there exists a Toeplitz sequence b over the same alphabet with h(b) ≤ 2h(a) and with the symmetric Cesàro average of |a-b| smaller than ε. The proof constructs b as a limit of an inductive sequence of overwritings: at stage M, blocks around multiples of a rapidly growing period l_M are replaced by the current central block, and the periods are chosen so that the symmetric average distance to a is bounded stage by stage. The entropy bound is obtained by comparing the number of l_M-words of a and b. As an application, the paper reduces Sarnak's conjecture to the statement that the Möbius function is disjoint from every zero-entropy Toeplitz sequence, using the Boyle–Downarowicz symbolic extension theorem to pass from arbitrary zero-entropy systems to subshifts.","tokens_in":5935,"tokens_out":33885,"duration_ms":339372,"significance":"If the main theorem is correct in its intended (limsup) form, this is a valuable reduction: the full Sarnak conjecture would follow from disjointness on a comparatively small and well-studied class of systems. The constructive approximation of arbitrary sequences by Toeplitz sequences with controlled entropy is interesting in its own right. The proof is essentially self-contained except for the standard symbolic extension theorem, and the inductive block construction is transparent and checkable. The paper is honest about relying on the deep Boyle–Downarowicz result. The central reduction survives a restatement from a limit assertion to a limsup assertion, so the significance is not diminished by the technical issue discussed below.","major_comments":[{"comment":"The proof does not establish the existence of the limit in Theorem 1.1(2). By summing the inequalities (II)_M one obtains, for each fixed N, that (1/(2N+1))Σ_{n=-N}^N |a_n-b_n| < Σ ε_n < ε. This yields at most limsup_{N→∞} of the symmetric average is ≤ ε; it does not imply that the limit exists. For an arbitrary sequence a, the symmetric Cesàro mean of |a_n-b_n| need not converge: on the positive-density subset of l1Z that is not overwritten by later stages, b is eventually equal to a0, so the average inherits the non-convergent Cesàro behavior of the subsequence a_{l1m}. The theorem, the abstract, and Corollary 1.2 should therefore be reformulated with 'limsup' (or, equivalently, with the stronger uniform-in-N bound that the proof actually gives). Corollary 1.2 remains valid in this limsup form, because the Möbius correlation difference is bounded by the average absolute difference and ε is arbitrary.","section":"Section 2, Eq. (5)"},{"comment":"Equation (5) asserts equality W_M(a(M′)) = W_M(a(M)) for all M′ ≥ M, but generally only the inclusion W_M(a(M′)) ⊆ W_M(a(M)) (equivalently, the cardinality inequality) is guaranteed. Later stages overwrite entire l_M-blocks with copies of central l_M-blocks, which can remove words from W_M without adding new ones; in particular, a word that appears only at an overwritten position is lost. The subsequent chain of inequalities only needs the inclusion/cardinality inequality, so the entropy estimate can be repaired by replacing this equality with the correct inclusion and adjusting the wording.","section":"Section 2, Eq. (5)"}],"minor_comments":[{"comment":"The symbol 'lim' should be replaced by 'limsup' (or the statement should say 'for every N') in both the abstract and Theorem 1.1(2). The same correction should be reflected in Corollary 1.2, where the reduction uses only the limsup version.","section":"Abstract and Theorem 1.1"},{"comment":"Property (II)_M is asserted without proof. Although the claimed bound is plausible and true, a short counting argument for the boundary terms (which motivate the factor 2kl_{M-1}/l_M) would significantly improve the clarity of the induction step.","section":"Section 2, paragraph before Step 1"},{"comment":"There are several typographical errors, for example 'regularly recurrent piont' in Section 1.4, 'if th e M¨obius' in the introduction, and a stray 'X_b' in the definition of [ϖ(M)] in Section 2. These should be corrected in revision.","section":"Throughout"},{"comment":"The reduction from the symmetric averages of Theorem 1.1 to the one-sided averages used in Sarnak's conjecture is not spelled out. A sentence explaining the standard factor-two argument would make the corollary fully self-contained.","section":"Section 1.5"}],"recommendation":"major_revision","confidential_remarks":"This is a solid constructive paper whose main reduction is valid after a localized correction. The limit/limsup overclaim is real and should be fixed, but it does not affect the Sarnak reduction. The false equality in Eq. (5) is also local and repairable; the entropy estimate can be justified with the correct inclusion. I recommend major revision rather than rejection, and I encourage the authors to restate the main theorem with limsup and to add the short counting argument for (II)_M."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing you should know: the headline theorem is not quite true as stated, but the fix is small and the application survives. The proof actually establishes limsup_N (1/(2N+1)) Σ |a_n − b_n| < ε, not the limit the abstract claims. The constructed Toeplitz sequence often has non-convergent Cesàro means for the error sequence: the first step pins the progression l1Z to a0, so the average contains a weighted average of |a_{l1m} − a0|, which for arbitrary a need not converge. So Theorem 1.1 should be restated with limsup. Corollary 1.2 only needs the limsup, since the Möbius correlation is bounded by the error average, so the advertised reduction to Toeplitz systems stands.\n\nWhat is genuinely new is the inductive block construction: any finite-alphabet sequence can be approximated by a Toeplitz sequence whose entropy is at most twice the original. The entropy estimate is the solid part. By looking at words on the growing periods l_M, the count of l_M-words in b is bounded by the count in a times a factor (l_M + 1), yielding h(b) ≤ 2h(a). The argument is self-contained modulo the Boyle–Downarowicz symbolic extension theorem, which is cited properly.\n\nSoft spots beyond the limsup issue are minor. Property (II)_j is stated as holding for all N, but for small N the boundary terms make it false; the proof only needs the bound for large N, so this is cosmetic. Equation (5) claims equality W_M(a(M′)) = W_M(a(M)), when the proof really shows containment and monotonicity of cardinality; the inequality used later remains valid. Neither affects the conclusion after restatement.\n\nBottom line: this is a serious paper with a fixable gap in the main statement. The reduction is a real contribution and the entropy technique is sound. It deserves a careful referee, who should insist on the limsup wording and check the small-N/boundary details.","headline":"The reduction to Toeplitz systems is a genuine contribution, but Theorem 1.1 is not proven as stated: the construction gives a limsup bound, not a limit, and the main corollary only needs the limsup.","tokens_in":6429,"tokens_out":20387,"would_cite":true,"duration_ms":202330,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B05","54H20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that any finite-alphabet sequence can be approximated in density by a Toeplitz sequence with entropy at most twice its own, and that Möbius disjointness for zero-entropy Toeplitz sequences would imply Sarnak's conjecture.","keywords":["Sarnak conjecture","Möbius function","Toeplitz systems","zero-entropy systems","topological entropy","symbolic dynamics","disjointness"],"falsifier":"Search for a zero-entropy Toeplitz sequence ${\\bf b}$ over $\\{1,\\ldots,k\\}$ for which $\\frac{1}{N}\\sum_{n=1}^{N}\\mu(n)b_n$ fails to converge to 0; any such sequence would violate the hypothesis of Corollary 1.2 and would also be a counterexample to Sarnak's conjecture. To test Theorem 1.1 directly, exhibit a sequence ${\\bf a}$ and an $\\varepsilon>0$ such that every Toeplitz sequence in the same alphabet either has entropy greater than $2h({\\bf a})$ or disagrees with ${\\bf a}$ on a set of density at least $\\varepsilon$.","tokens_in":5503,"feed_emoji":"🔢","tokens_out":13685,"duration_ms":125578,"temperature":0.7,"pith_summary":"Every two-sided sequence over a finite alphabet can be matched, except on an arbitrarily sparse set of positions, by a Toeplitz sequence in the same alphabet whose topological entropy is no more than twice the original sequence's entropy. A Toeplitz sequence is one in which every coordinate is periodic, possibly with a different period; its orbit closure is a zero-entropy minimal symbolic system that is an almost one-to-one extension of an odometer. The proof constructs the approximating sequence by repeatedly periodizing a central block on a nested family of lattices, pushing the changed positions out to vanishing density. Because the approximation error is controlled in Cesàro mean, Möbius disjointness transfers from the approximating Toeplitz sequence back to any zero-entropy sequence. Together with a known symbolic-extension theorem, this reduces Sarnak's conjecture to the single question of Möbius disjointness from zero-entropy Toeplitz sequences.","feed_headline":"Sarnak conjecture reduces to zero-entropy Toeplitz systems","feed_subtitle":"If the Möbius function decorrelates from every zero-entropy Toeplitz sequence, Sarnak's conjecture follows.","key_machinery":"The mechanism carrying the proof is a block-periodization recursion on a full shift, producing a Toeplitz sequence — one in which each coordinate is constant on some arithmetic progression $n+p_n\\mathbb{Z}$ — as the limit of auxiliary sequences ${\\bf a}^{(M)}$. At stage $M$, the central word $\\varpi^{(M)}={\\bf a}^{(M-1)}[-l_{M-1},l_{M-1}-1]$ is copied onto every block $rl_M-l_{M-1},\\ldots,rl_M+l_{M-1}-1$, where $l_M$ is a multiple of $l_{M-1}$ chosen so that $2kl_{M-1}/l_M<\\varepsilon_M$; all other positions keep the previous symbols. The per-stage density errors (II)$_M$ sum below $\\varepsilon$, and each coordinate is periodic from some stage on, so the limit is Toeplitz. Entropy is controlled by the grid word-count inequality ${\\#}B_{l_M}({\\bf b})\\le (l_M+1)({\\#}W_M({\\bf b}))^2\\le (l_M+1)({\\#}B_{l_M}({\\bf a}))^2$, which after dividing by $l_M$ and taking limits yields $h({\\bf b})\\le 2h({\\bf a})$.","core_discovery":"The central claim is Theorem 1.1: for any ${\\bf a}=(a_n)_{n\\in\\mathbb{Z}}\\in\\{1,\\ldots,k\\}^{\\mathbb{Z}}$ and any $\\varepsilon>0$, there exists a Toeplitz sequence ${\\bf b}\\in\\{1,\\ldots,k\\}^{\\mathbb{Z}}$ such that $h({\\bf b})\\le 2h({\\bf a})$ and $\\lim_{N\\to\\infty}\\frac{1}{2N+1}\\sum_{n=-N}^{N}|a_n-b_n|<\\varepsilon$. The proof defines ${\\bf a}^{(M)}$ inductively: the central word ${\\bf a}^{(M-1)}[-l_{M-1},l_{M-1}-1]$ is copied onto every translate of the lattice $l_M\\mathbb{Z}$, with $l_{M-1}\\mid l_M$ and with $l_M$ so large that the positions changed at stage $M$ have density below $\\varepsilon_M$, where $\\sum_M\\varepsilon_M<\\varepsilon$. The limit ${\\bf b}=\\lim_M{\\bf a}^{(M)}$ is Toeplitz because every finite central word eventually recurs with period $l_M$, and the grid word-count inequalities give $h({\\bf b})\\le 2h({\\bf a})$. If ${\\bf a}$ has zero entropy, so does ${\\bf b}$. Hence, using the symbolic-extension theorem for zero-entropy systems, the authors obtain Corollary 1.2: Möbius disjointness from every zero-entropy Toeplitz sequence implies Sarnak's conjecture.","pith_inferences":["The factor 2 in the entropy bound appears as an artifact of counting $l_M$-blocks of ${\\bf b}$ through pairs of grid blocks; a refined count might lower it to $1+\\delta$, though the paper makes no such claim.","The construction makes Toeplitz sequences dense in the Cesàro-mean metric on every full shift under a bounded entropy penalty, so Toeplitz systems could serve as a test family for other multiplicative-function disjointness problems.","If any zero-entropy Toeplitz sequence ever fails Möbius disjointness, the reduction would convert that single counterexample into a counterexample to Sarnak for arbitrary deterministic systems, identifying Toeplitz systems as the critical case.","A similar nested periodization may adapt to higher-dimensional actions or to approximation of non-symbolic systems, though the paper treats only two-sided one-dimensional sequences."],"forward_implications":["Sarnak's conjecture is settled as soon as the Möbius function is shown to be linearly disjoint from every zero-entropy Toeplitz sequence.","For any zero-entropy sequence ${\\bf a}$, the constructed approximant ${\\bf b}$ also has zero entropy, so deterministic correlations with $\\mu$ need only be controlled on Toeplitz systems.","The approximation is quantitative: the difference between the Möbius correlation of ${\\bf a}$ and that of ${\\bf b}$ is bounded by the Cesàro density of the positions where they disagree, which is smaller than $\\varepsilon$.","The construction applies to all finite-alphabet sequences, not just zero-entropy ones, and bounds entropy inflation by a factor of 2.","Once Toeplitz disjointness is proved, the symbolic-extension theorem automatically extends Sarnak from subshifts to all zero-entropy topological dynamical systems."],"supporting_citations":[{"why":"Supplies the symbolic-extension theorem used to pass from arbitrary zero-entropy systems to zero-entropy subshifts, the entry point for the approximation theorem.","marker":"[1]"},{"why":"Provides the characterization of Toeplitz systems as almost one-to-one extensions of odometers, fixing the target class for the construction.","marker":"[2]"},{"why":"Gives the regularly recurrent point characterization underlying the fact that the constructed limit sequence is Toeplitz.","marker":"[4]"},{"why":"Formulates the Möbius linear-disjointness conjecture that the paper reduces to the zero-entropy Toeplitz case.","marker":"[5]"}],"fun_headline_variants":["Sarnak conjecture narrows to Toeplitz systems","Toeplitz systems key to Sarnak conjecture","Zero-entropy Toeplitz sequences decide Sarnak","Reducing Sarnak to Toeplitz entropy bound","Sarnak conjecture hinges on Toeplitz zero entropy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole reduction to sequences rests on the external theorem that every zero-entropy topological dynamical system has a zero-entropy subshift extension; without it, the argument would cover only zero-entropy sequences whose orbit closures are already subshifts.","fun_headline_variants_meta":{"raw":{"variants":["Sarnak conjecture narrows to Toeplitz systems","Toeplitz systems key to Sarnak conjecture","Zero-entropy Toeplitz sequences decide Sarnak","Reducing Sarnak to Toeplitz entropy bound","Sarnak conjecture hinges on Toeplitz zero entropy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000228,"raw_usage":{"total_tokens":1520,"prompt_tokens":1038,"completion_tokens":482,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":654,"completion_tokens_details":{"reasoning_tokens":397}},"tokens_in":654,"tokens_out":482,"duration_ms":4713,"temperature":1.0,"reasoning_tokens":397,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:07:25.041304+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a zero-entropy Toeplitz sequence ${\\bf b}$ over $\\{1,\\ldots,k\\}$ for which $\\frac{1}{N}\\sum_{n=1}^{N}\\mu(n)b_n$ fails to converge to 0; any such sequence would violate the hypothesis of Corollary 1.2 and would also be a counterexample to Sarnak's conjecture. To test Theorem 1.1 directly, exhibit a sequence ${\\bf a}$ and an $\\varepsilon>0$ such that every Toeplitz sequence in the same alphabet either has entropy greater than $2h({\\bf a})$ or disagrees with ${\\bf a}$ on a set of density at least $\\varepsilon$.","supporting_citations":[{"cited_title":"The entropy theory of symbolic extensions","cited_arxiv_id":null,"evidence_quote":"Supplies the symbolic-extension theorem used to pass from arbitrary zero-entropy systems to zero-entropy subshifts, the entry point for the approximation theorem."},{"cited_title":"Algebraic and topological dynamics, 7–37, Contemp","cited_arxiv_id":null,"evidence_quote":"Provides the characterization of Toeplitz systems as almost one-to-one extensions of odometers, fixing the target class for the construction."},{"cited_title":"Dynamical systems disjoint from any minimal system","cited_arxiv_id":null,"evidence_quote":"Gives the regularly recurrent point characterization underlying the fact that the constructed limit sequence is Toeplitz."},{"cited_title":"Three lectures on the M ¨obius function, randomness and dynamics","cited_arxiv_id":null,"evidence_quote":"Formulates the Möbius linear-disjointness conjecture that the paper reduces to the zero-entropy Toeplitz case."}],"review_version":1}