{"id":"c4c0600e-3979-4054-9cdf-82227b67687e","arxiv_id":"2505.09253","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Besicovitch covering-number scaling of generic orbits is a block-code invariant that distinguishes systems with identical discrete spectra, as shown for B-free shifts and golden-rotation codings.","lead":"Kasjan and Keller study Besicovitch covering numbers of individual symbolic orbits and show their growth rate is an invariant under sliding-block codes. They construct continuous families of sequences with identical spectrum and orbit closure but different growth rates, so the sequences cannot be related by finite block codes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing point is the proof of (P3): the two-sided inclusion (4.6) rests on unstated geometric facts about the W_n jumps; this is checkable and correctable but not yet fully pinned down.","rationale":"The reader's weakest assumption correctly identifies property (P3) as the load-bearing step. My reading agrees: Proposition 4.1c reduces the whole non-block-code-equivalence conclusion to the scaling in (P3), and the proof of (P3) rests on inclusions (4.5)-(4.7), especially the geometric lower bound behind (4.6). The concern is not that the construction is known to fail; rather, the most delicate estimate is asserted with 'easily seen' and depends on a missing diagram and an unproved inclusion in (O1). This is the least externally supported step in the new construction, and it is exactly where an exponent error would change the final result. The mis-stated Eq. (4.2) is real but appears correctable: the coding holds on the appropriate half-interval, and the complementary half gives the complement of ω, whose orbit is also dense. Other parts of the paper, notably the general theory of Theorem 2.10 and the B-free computations in Sections 3.2-3.5, are substantially independent and appear sound. Since the identified concern is a verification gap rather than a demonstrated contradiction, and since the reader already made the verdict conditional, I recommend keeping the conditional verdict rather than accepting unconditionally or rejecting.","tokens_in":34847,"tokens_out":22269,"duration_ms":214490,"concrete_test":"Fix s=2 and s=3; generate W_n for n=1,...,15 using (4.1) explicitly as finite unions of intervals with exact arithmetic. Verify (a) that λ(W_n△(W_n+h))≥c·|h|q_{n-1} for h in I_n with a constant c uniform in n, and (b) that the empirical measure of {h: λ(W_n△(W_n+h))<ε} over ε∈(ε_{n+1},ε_n] is bracketed by C·ε^{s/(s-1)} with a uniform C. A deviation of more than a few percent in the empirical exponent would indicate that the proof of (P3) is incomplete or incorrect; an analytic re-derivation of (O1) with explicit constants for all s>1 would settle it definitively.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that 1<s<s' implies no block-code equivalence is carried entirely by Proposition 4.1c, which converts Besicovitch ball measure into the window-translation scaling (P3) via Theorem 2.10c. The proof of (P3) is the chain (4.4)-(4.7). The most delicate link is the upper inclusion in (4.6), H_n⊆I_{n-L}: it uses the lower bound λ(W_n△(W_n+h))≥|h|q_{n-1} for all h∈I_n, justified by the assertion that 1_{W_n} has a jump at each c_k (q_n≤k<q_{n+1}) and is constant on intervals of length at least δ_n. This is plausible from the recursive construction (4.1), but it is not proved with the uniformity needed, and the constants entering (4.4) also depend on the unproved inclusion [-2δ_n,2δ_n]⊂J_{[sn]} in (O1). If either geometric fact fails for some s>1, the ε^{s/(s-1)} scaling could be off and the non-equivalence conclusion would be unsupported. The displayed Eq. (4.2) is additionally mis-stated: it holds on the half-interval corresponding to ω_{k-q_n}, not on all of I_n. This appears repairable via the complement's dense orbit, but it compounds the need to verify the geometric estimates explicitly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Besicovitch covering numbers N_epsilon(O(x)) of individual shift orbits and proposes them as invariants for block code equivalence. The main general result, Theorem 2.10c, gives two-sided bounds for these covering numbers in terms of the mu-measure of Besicovitch balls, under the assumption that the ambient measure is ergodic with discrete spectrum. The authors then specialize to B-free systems, deriving explicit arithmetic identities and estimates that yield amorphic complexities for several Toeplitz examples and for the square-free shift, where the critical exponent on the exponential scale is identified as 1. The final section constructs, for each s>1, a Borel window W_s for the golden rotation such that the corresponding Mirsky measures all have the same discrete spectrum generated by one eigenvalue, yet Lebesgue-a.e. point in the associated full shift has amorphic complexity s/(s-1). Consequently, for 1<s<s', a.e. pair of points is not block code equivalent, despite identical spectra and full orbit closures.","tokens_in":35122,"tokens_out":12877,"duration_ms":116911,"significance":"If the construction in Section 4.3 can be made fully rigorous, the paper is significant. It introduces a genuinely new invariant for block code equivalence that is finer than spectrum or topological orbit-closure isomorphism in the examples: two systems can have identical spectrum and identical full-shift orbit closure while their typical points have different amorphic complexity. The general bounds in Theorem 2.10c are clean and likely to be a useful tool beyond the examples treated. The arithmetic work on B-free systems is explicit and checkable, and the square-free example provides a concrete computation of covering-number scaling on a non-polynomial scale. The paper is clearly written in its general parts and the main ideas are well motivated.","major_comments":[{"comment":"The displayed equivalence (4.2), 'for all x in I_n and all k in [q_n, q_{n+1}): R_k x in W_n iff omega_{k-q_n}=1', is not correct for all x in I_n. For x in I^0_n = (0, delta_n] the correct equivalence is with omega_{k-q_n}=0 instead of 1. The subsequent proof of (P1) only uses the statement on the half-interval I^1_n, so the error is repairable, but the displayed equivalence and the text around it should be rewritten to state the correct half-interval version and to explain how the full-measure statement is derived from it.","section":"Section 4.3, Eq. (4.2)"},{"comment":"The right inclusion H_n subseteq I_{n-L} in (4.6) is the most delicate step in the proof of (P3). It relies on the unproved lower bound lambda(W_n triangle (W_n+h)) >= |h| q_{n-1} for h in I_n. This bound requires that the indicator 1_{W_n} has a jump at each c_k for q_n <= k < q_{n+1} and is constant on both sides on intervals of length at least delta_n, and that no contributions from earlier stages W_m with m<n interfere. A complete proof must use the disjointness provided by (O1) and control the interaction between the intervals of different stages. As written, the estimate is asserted rather than verified, and it is load-bearing for the epsilon^{s/(s-1)} scaling in (4.7), hence for Proposition 4.1c and the main non-equivalence conclusion.","section":"Section 4.3, Eq. (4.6)"},{"comment":"The inclusion [-2 delta_n, 2 delta_n] subset J_[sn] is stated as immediate 'by definition of delta_n'. This inclusion is used to conclude |k-k'| >= q_[sn] in (O1) and in the bound on #T_{n,n'} in (O2), and it determines the constants in (4.4). It should be proved explicitly from the definitions of delta_n and J_m; the argument is short, but it cannot be omitted from a construction that carries the paper's main theorem.","section":"Section 4.3, (O1)"}],"minor_comments":[{"comment":"The half-open conventions for I^1_n, I^0_n and I_n should be stated once in a precise way; the current text and the footnote use different notations (e.g. I^+_n and I^-_n), which makes the quantifier in (4.2) harder to track.","section":"Section 4.3"},{"comment":"The sentence 'the Symbols like dN, dN etc. have analogous meaning' contains capitalization and notation inconsistencies; please harmonize the density notation throughout the paper.","section":"Section 5.1"},{"comment":"The assertion that epsilon_{ell_n}/epsilon_{ell_{n+1}} -> 1 is used without proof; it follows from the asymptotic formula for epsilon_{ell_n}, but the derivation should be shown.","section":"Example 3.18"},{"comment":"The abstract states that the covering numbers are 'sufficiently different' for the two parameters, while the body gives the exact amorphic complexity; the abstract should be made consistent and quantitative.","section":"Abstract and Section 1.4"},{"comment":"The geometric fact quoted from [4, p.28] is used in (O2); a precise lemma or theorem number from [4] would help the reader verify the claim.","section":"Section 4.2, (GR6)"}],"recommendation":"major_revision","confidential_remarks":"The central construction in Section 4.3 is the heart of the paper's main new example, and the current level of detail is too compressed for the load-bearing geometric estimates. I found no fundamental contradiction, and the gaps appear repairable, so I would not recommend rejection. If the authors provide complete proofs of (O1) and (4.6) and fix the quantifier in (4.2), the paper should be publishable in a strong dynamical systems journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the paper. The headline is that the individual-orbit Besicovitch covering numbers are a genuinely useful block-code invariant, and the two example families—B-free shifts and the golden-rotation windows—make the case. Theorem 2.10c is the engine: N_epsilon(O(x)) is sandwiched between inverse measures of d1-balls. That is clean and new in this form. The applications are substantial: exact formulas for d1(eta,sigma^r eta) for pairwise coprime B, a Toeplitz family with pairwise distinct amorphic complexities, and the construction of measures mu_s with identical spectrum but amorphic complexity s/(s-1) for typical orbits. I think the central claims are true.\n\nWhat is new: the individual-orbit viewpoint on covering numbers, the arithmetic formulas in Cor 3.10-3.12, and especially the golden-rotation family. The general equivalence theorems partly repackage known material from Lenz-Spindeler-Strungaru and Wiener-Wintner, but the applications are new and concrete.\n\nWhere the paper is soft: Eq (4.2) is overbroad. It is not true for all x in I_n; the correct statement is that membership of R^k x in W_n equals omega_{k-q_n} when x in I^1_n and equals 1-omega_{k-q_n} when x in I^0_n. The proof of (P1) only needs the I^1_n version, so this is repairable, but the displayed claim is false as written.\n\nThe bigger soft spot is the proof of (P3), specifically the inclusions (4.6). The bound lambda(W_n Delta (W_n+h)) >= |h| q_{n-1} for h in I_n is plausible—each of the q_{n-1} jumps contributes—but the uniformity is not fully proved. Same for the upper bound 4 q_n |h| and the interval-disjointness facts behind (O1). I checked the orders of magnitude: with delta_n approximately alpha^{s n} and q_n approximately alpha^{-n}, both inclusions work for L large enough, and [-2 delta_n, 2 delta_n] subset J_{[sn]} holds because 2 alpha^{s+2} < 1. So (P3) is probably correct, but a referee should demand a complete, constant-explicit proof of (4.6).\n\nCitations: some supporting facts come from the authors' own prior work, but they are used appropriately and the main line does not depend on unverified self-citation. The paper is honest about what is synthesized.\n\nAudience: symbolic dynamics people working on B-free shifts, amorphic complexity, or block-code invariants. It deserves a serious referee; with the quantifier fix and a fuller proof of (4.6), I would be happy to see it accepted. If I were the editor, I would send it out.","headline":"Genuinely new invariant and solid applications; two repairable gaps in the golden-rotation section, not fatal.","tokens_in":35694,"tokens_out":8739,"would_cite":true,"duration_ms":82985,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A35","37A44"],"pacs":[],"model":"deepseek-v4-flash","headline":"Besicovitch covering numbers of individual orbits are a block-code invariant, and golden-rotation codings show this invariant separates systems with identical spectra.","keywords":["Besicovitch pseudo-metric","covering numbers","amorphic complexity","block code equivalence","B-free shifts","weak model sets","discrete spectrum","golden rotation"],"falsifier":"For a fixed $s>1$, approximate $W_s$ by the finite-stage sets $W_n$ and numerically estimate the tail $\\lambda\\{h\\in\\mathbb T: \\lambda(W_s\\triangle(W_s+h))\\le \\epsilon\\}$ for shrinking $\\epsilon$; property (P3) predicts a clean $\\epsilon^{s/(s-1)}$ power law, so a systematic deviation would invalidate Proposition 4.1c. Alternatively, check the quantifier in Eq. (4.2): the coding property is proved on the half-interval $I_n^1$, while the text states it for all $x\\in I_n$; if the full-interval statement is needed for property (P1), the proof must be corrected before the conclusion is fully established.","tokens_in":34619,"feed_emoji":"🔄","tokens_out":14424,"duration_ms":126349,"temperature":0.7,"pith_summary":"This paper establishes that the Besicovitch covering numbers of individual shift orbits—the count of open balls of radius $\\epsilon$ needed to cover the orbit in the Besicovitch pseudo-metric $d_1$—form an invariant for block-code equivalence, and that this invariant can distinguish systems that have identical spectra and identical orbit closures. The engine is an inequality (Theorem 2.10c) bounding these covering numbers between the reciprocals of the measure of Besicovitch balls of radius $\\epsilon$ and $\\epsilon/2$, valid for every point generic for an ergodic discrete-spectrum measure. On this basis the paper computes or bounds covering numbers for $\\mathcal B$-free shifts, including structured examples with mutually distinct amorphic complexities and a square-free shift whose covering numbers grow faster than any power. It then constructs a one-parameter family of codings of the golden rotation with windows $W_s$, $s>1$, whose typical sequences have amorphic complexity $s/(s-1)$; for $s\\neq s'$ those sequences are not block-code equivalent, although all the measures share the same pure point spectrum generated by a single rotation. A reader would care because this supplies a concrete, orbit-wise computable scaling invariant for zero-entropy symbolic systems that classical spectral invariants see as identical.","feed_headline":"Covering numbers tell spectrally identical shifts apart","feed_subtitle":"Besicovitch ball counts give a block-code invariant; golden-rotation windows realize every amorphic complexity s/(s-1).","key_machinery":"The load-bearing object is the Besicovitch pseudo-metric $d_1(x,y)=\\limsup_{n\\to\\infty}\\frac{1}{2n+1}\\#\\{|i|\\le n: x_i\\neq y_i\\}$ on $A^{\\mathbb Z}$ and the orbit closure $O(\\tilde x)$ inside the quotient Besicovitch space. For an ergodic discrete-spectrum measure $\\mu$, all $\\mu$-generic points lie in one $D_1$-equivalence class $X_\\mu$, all are Besicovitch almost periodic, and $O(\\tilde x)$ is a compact abelian group carrying the unique invariant measure. The quantitative link is Theorem 2.10c: $\\mu(B_{d_1}(x,\\epsilon))^{-1} \\leq N_\\epsilon(O(x)) \\leq \\mu(B_{d_1}(x,\\epsilon/2))^{-1}$, which turns the scaling of covering numbers into the scaling of the $\\mu$-measure of Besicovitch balls. The second main mechanism is the window construction $W_s$ for the golden rotation in Section 4.3, built from nested intervals $V_n^0$, $V_n^1$ along the return times $q_n$ with $q_{n+1}=q_n+q_{n-1}$, with the defining property that $\\lambda\\{h\\in\\mathbb T: \\lambda(W_s\\triangle(W_s+h))\\le \\epsilon\\} \\asymp \\epsilon^{s/(s-1)}$; this tail scaling is exactly what feeds the ball-measure estimate and yields amorphic complexity $s/(s-1)$.","core_discovery":"The paper's central claim is that for an ergodic shift-invariant measure $\\mu$ with discrete spectrum, the covering-number function $N_\\epsilon(O(x))$ of any $\\mu$-generic Besicovitch almost periodic point $x$ is finite for every $\\epsilon>0$, is independent of the chosen generic point, and satisfies $\\mu(B_{d_1}(x,\\epsilon))^{-1} \\leq N_\\epsilon(O(x)) \\leq \\mu(B_{d_1}(x,\\epsilon/2))^{-1}$. Because a sliding block code is Lipschitz with respect to $d_1$, the equivalence class of this covering-number function is preserved under block-code equivalence, so the amorphic complexity of an orbit—the polynomial growth rate of its covering numbers—is a block-code invariant. The paper then uses this invariant to separate systems that are spectrally identical: for the golden rotation and the window family $W_s$ satisfying property (P3), the measures $\\mu_s$ all have spectrum $\\{e^{2\\pi i \\ell \\alpha}:\\ell\\in\\mathbb Z\\}$ and almost every orbit is dense in the full shift $\\{0,1\\}^{\\mathbb Z}$, yet the amorphic complexity is $s/(s-1)$. Hence for $1<s<s'$ the typical orbits are not block-code equivalent. For $\\mathcal B$-free shifts, the paper develops arithmetic identities expressing $d_1(\\eta,\\sigma^r\\eta)$ as the density of a symmetric difference of multiples sets, and uses them to estimate covering numbers for structured and square-free cases.","pith_inferences":["Because the invariant survives factor maps whose code length has finite moments of all orders, the same golden-rotation family should separate orbits under a broader class of maps than finite block codes; this is a testable robustness property of the invariant.","The window construction uses the golden rotation; a natural extension is to replace $\\alpha$ by other badly approximable rotations, where the symmetric-difference tail would presumably scale with a different exponent, producing further continuous families of non-equivalent systems.","The relation $N_\\epsilon(O(x))\\approx \\mu(B_{d_1}(x,\\epsilon))^{-1}$ suggests that the covering-number dimension of a weak model set could be read off from the scaling of its autocorrelation or window radial distribution, connecting this invariant to diffraction-style computations.","Footnote 22 indicates that choices made inside each pair of half-intervals can move the coded point $\\varphi_s(0)$ outside $X_\\mu$; if so, the covering-number dimension may depend on the representative within a $D_1$-equivalence class, which would be worth checking explicitly."],"forward_implications":["Amorphic complexity, and more generally the equivalence class of the covering-number function, is a genuine block-code invariant that can be computed orbit-wise for generic points, not only for entire mean-equicontinuous subshifts.","The golden-rotation family yields a continuous family of measures $\\mu_s$, all with the same pure point spectrum and with almost every orbit dense in $\\{0,1\\}^{\\mathbb Z}$, whose typical points are pairwise non-block-code equivalent for different values of $s$.","For $\\mathcal B$-free shifts, the identity $d_1(\\eta,\\sigma^r\\eta)=d(M_{\\mathcal B}\\triangle(r+M_{\\mathcal B}))$ gives explicit upper and lower bounds on covering numbers in terms of least common multiples, enough to separate structured $\\mathcal B$-free shifts by their amorphic complexity while their maximal equicontinuous factors are isomorphic.","The square-free shift has infinite amorphic complexity, but its covering numbers still have a finite critical exponent on the power-exponential scale $\\epsilon\\mapsto \\exp(-\\epsilon^{-\\alpha})$, namely $\\alpha=1$, refining the picture for this classical system."],"supporting_citations":[{"why":"Supplies the characterization of mean and Besicovitch almost periodic points and the discrete-spectrum equivalence used in Theorem 2.9.","marker":"[27]"},{"why":"Classical 1941 almost-periodicity theorem underlying the discrete-spectrum versus Besicovitch almost periodicity equivalence.","marker":"[36]"},{"why":"Establishes the invariant measure, odometer group, and genericity of $\\mathcal B$-free indicator sequences used throughout Section 3.","marker":"[10]"},{"why":"Introduces amorphic complexity and the covering/spanning number viewpoint on which the new invariant is built.","marker":"[11]"},{"why":"Gives the finite-covering-numbers criterion for mean equicontinuity and the amorphic complexity framework for group actions.","marker":"[12]"},{"why":"Provides the window representation of $\\mathcal B$-free sets, including tautness and Haar aperiodicity used in Corollary 3.3.","marker":"[19]"},{"why":"Determines the spectrum of weak model sets with Borel windows, used in Proposition 4.1a.","marker":"[24]"},{"why":"Provides the isomorphism theorem for ergodic rotations used to identify the compact abelian group structure of orbit closures.","marker":"[35]"}],"fun_headline_variants":["Covering numbers split shifts with identical spectra","Amorphic complexity: block-code invariant beyond spectrum","Besicovitch ball counts tell spectrally twin shifts apart","Orbit covering numbers separate same-spectrum systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the nested-interval window $W_s$ built in Section 4.3 really has the advertised symmetric-difference tail: the measure of the set of shifts $h$ for which $W_s$ differs from $W_s+h$ by at most $\\epsilon$ grows like $\\epsilon^{s/(s-1)}$; if that estimate fails, the amorphic complexity $s/(s-1)$ and the non-block-code-equivalence conclusion collapse.","fun_headline_variants_meta":{"raw":{"variants":["Covering numbers split shifts with identical spectra","Amorphic complexity: block-code invariant beyond spectrum","Besicovitch ball counts tell spectrally twin shifts apart","Orbit covering numbers separate same-spectrum systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000272,"raw_usage":{"total_tokens":1721,"prompt_tokens":1124,"completion_tokens":597,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":740,"completion_tokens_details":{"reasoning_tokens":536}},"tokens_in":740,"tokens_out":597,"duration_ms":6313,"temperature":1.0,"reasoning_tokens":536,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:37:42.380019+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed $s>1$, approximate $W_s$ by the finite-stage sets $W_n$ and numerically estimate the tail $\\lambda\\{h\\in\\mathbb T: \\lambda(W_s\\triangle(W_s+h))\\le \\epsilon\\}$ for shrinking $\\epsilon$; property (P3) predicts a clean $\\epsilon^{s/(s-1)}$ power law, so a systematic deviation would invalidate Proposition 4.1c. Alternatively, check the quantifier in Eq. (4.2): the coding property is proved on the half-interval $I_n^1$, while the text states it for all $x\\in I_n$; if the full-interval statement is needed for property (P1), the proof must be corrected before the conclusion is fully established.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the characterization of mean and Besicovitch almost periodic points and the discrete-spectrum equivalence used in Theorem 2.9."},{"cited_title":"Wiener and A","cited_arxiv_id":null,"evidence_quote":"Classical 1941 almost-periodicity theorem underlying the discrete-spectrum versus Besicovitch almost periodicity equivalence."},{"cited_title":"Dymek, S","cited_arxiv_id":null,"evidence_quote":"Establishes the invariant measure, odometer group, and genericity of $\\mathcal B$-free indicator sequences used throughout Section 3."},{"cited_title":"Fuhrmann, M","cited_arxiv_id":null,"evidence_quote":"Introduces amorphic complexity and the covering/spanning number viewpoint on which the new invariant is built."},{"cited_title":"Fuhrmann, M","cited_arxiv_id":null,"evidence_quote":"Gives the finite-covering-numbers criterion for mean equicontinuity and the amorphic complexity framework for group actions."},{"cited_title":"Kasjan, G","cited_arxiv_id":null,"evidence_quote":"Provides the window representation of $\\mathcal B$-free sets, including tautness and Haar aperiodicity used in Corollary 3.3."},{"cited_title":"Keller, C","cited_arxiv_id":null,"evidence_quote":"Determines the spectrum of weak model sets with Borel windows, used in Proposition 4.1a."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the isomorphism theorem for ergodic rotations used to identify the compact abelian group structure of orbit closures."}],"review_version":1}