REVIEW 2 major objections 4 minor 5 references
Reducing the Sarnak Conjecture to Toeplitz systems
T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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 $h({\bf b})\le 2h({\bf a})$.
What would settle it
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 $\varepsilon$.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Section 2, Eq. (5)] 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 2, Eq. (5)] 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.
minor comments (4)
- [Abstract and Theorem 1.1] 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 2, paragraph before Step 1] 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.
- [Throughout] 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 1.5] 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.
Circularity Check
No circularity: Theorem 1.1 is an explicit construction and the Sarnak reduction is a genuine conditional implication.
full rationale
The paper's central claim is an approximation theorem: for every finite-alphabet bi-infinite sequence a and every epsilon>0, it constructs a Toeplitz sequence b with h(b) <= 2h(a) and Cesaro mean |a-b| < epsilon. The construction in Section 2 explicitly builds b as the limit of a^{(n)} by periodic overwriting on nested lattices l_n Z. The closeness estimate (II)_n is proved by counting overwritten positions, and the entropy bound follows from the block-counting inequalities (5)-(7) together with #W_M(b) <= #W_M(a). No step fits a parameter to the target Sarnak correlation, no assumption of Mobius disjointness or the Sarnak conjecture is used, and the reduction in Corollary 1.2 is purely conditional: it says that if the hypothesis about Toeplitz systems holds, then Sarnak holds, which is exactly the intended logical implication. The only external load-bearing input is the Boyle-Downarowicz symbolic extension theorem [1, Theorem 8.6], which is an independent published result and not a self-citation; the other citations, including the authors' own [4], are used for standard characterizations of regularly recurrent points and Toeplitz systems rather than as premises of the main reduction. A possible issue that the limit in Theorem 1.1(2) may require a limsup restatement is a correctness concern, not circularity, because the approximation construction itself is independent of the target conjecture. Accordingly no circular step is present.
Assumptions & free parameters
free parameters (2)
- epsilon_M (error tolerances) =
chosen inductively with ε_M < (1/2)ε_{M-1} and Σ ε_M < ε
- l_M (period scales) =
chosen inductively with l_{M-1} | l_M, l_M → ∞, and 2k l_{M-1}/l_M ≤ ε_M
assumptions (4)
- domain assumption Boyle-Downarowicz symbolic extension theorem: any zero-entropy t.d.s. has a zero-entropy subshift extension
- domain assumption Toeplitz systems are exactly orbit closures of regularly recurrent points, equivalently almost 1-1 symbolic extensions of adding machines
- standard math Topological entropy of a subshift is the limit of (1/n) log #B_n(X), and subadditivity ensures the limit exists
- standard math Cesàro averages of bounded sequences can be compared via limsup; finite initial segments are negligible
Cite this review
Pith. "Pith review of Reducing the Sarnak Conjecture to Toeplitz systems." pith.science (2026). https://pith.science/paper/XO4X5MAS
@misc{pith2026190807554,
author = {Pith},
title = {Pith review of: Reducing the Sarnak Conjecture to Toeplitz systems},
year = {2026},
howpublished = {\url{https://pith.science/paper/XO4X5MAS}},
note = {Machine review of arXiv:1908.07554}
}
abstract
In this paper, we show that for any sequence ${\bf a}=(a_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ and any $\epsilon>0$, there exists a Toeplitz sequence ${\bf b}=(b_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ such that the entropy $h({\bf b})\leq 2 h({\bf a})$ and $\lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N|a_n-b_n|<\epsilon$. As an application of this result, we reduce Sarnak Conjecture to Toeplitz systems, that is, if the M\"{o}bius function is disjoint from any Toeplitz sequence with zero entropy, then the Sarnak conjecture holds.
Reference graph
Works this paper leans on
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[2]
Algebraic and topological dynamics, 7–37, Contemp
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Tomasz Downarowicz, Stanislaw Kasjan, Odometers and T oeplitz systems revisited in the context of Sarnak’s conjecture, Studia Mathematica, 229, 45-72, 2015
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Dynamical systems disjoint from any minimal system
Wen Huang and Xiangdong Y e. Dynamical systems disjoint from any minimal system . Trans. Amer. Math., 357(2):669–694, 2005
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[5]
Three lectures on the M ¨obius function, randomness and dynamics
Peter Sarnak. Three lectures on the M ¨obius function, randomness and dynamics . http://publications.ias.edu/sarnak/. WU WEN -T SUN KEY LABORATORY OF MATHEMATICS , USTC, C HINESE ACADEMY OF SCIENCES , DEPARTMENT OF MATHEMATICS , U NIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA , H EFEI , ANHUI , 230026, P.R. C HINA AND INSTITUTE OF MATHEMATICS , P OLISH ACA...
Reviewed August 14, 2026 · model on record in the stance chip above.
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