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REVIEW 3 major objections 5 minor 36 references

Besicovitch covering numbers for $\mathcal B$-free and other shifts

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Besicovitch covering numbers of individual orbits are a block-code invariant, and golden-rotation codings show this invariant separates systems with identical spectra.

desk verdict Genuinely new invariant and solid applications; two repairable gaps in the golden-rotation section, not fatal. read the letter →

arxiv 2505.09253 v2 pith:NFQJJTZN submitted 2025-05-14 math.DS

Stanis{l}aw Kasjan , Gerhard Keller This is my paper · ORCID
classification math.DS MSC 37A3537A44
keywords Besicovitchpseudo-metriccoveringnumbersamorphiccomplexityblockcodeequivalenceB-freeshiftsweakmodelsetsdiscretespectrumgoldenrotation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $s/(s-1)$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Section 4.3, Eq. (4.2)] 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.
  2. [Section 4.3, Eq. (4.6)] 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.
  3. [Section 4.3, (O1)] 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.
minor comments (5)
  1. [Section 4.3] 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.
  2. [Section 5.1] The sentence 'the Symbols like dN, dN etc. have analogous meaning' contains capitalization and notation inconsistencies; please harmonize the density notation throughout the paper.
  3. [Example 3.18] 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.
  4. [Abstract and Section 1.4] 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.
  5. [Section 4.2, (GR6)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Besicovitch covering-number estimates are proved from ball-measure bounds and an explicit window construction, with only background self-citations.

full rationale

The paper's main claims do not reduce to their inputs. Theorem 2.10c (Eq. 2.5) is proved by a Frostman-type argument: the lower bound follows from an invariant measure on the isometric orbit closure and Lemma 2.5, and the upper bound from maximal separated sets, so covering numbers are bounded by ball measures, not defined to be them. The B-free estimates (Lemmas 3.13, 3.15 and Example 3.16) are derived from arithmetic identities proved in Sections 3.2 and 5.2 plus the general ball-measure bound. The pivotal golden-rotation example is constructed, not fitted: the recursive window definition (4.1) and the Fibonacci geometry are used to prove the two-sided inclusions (4.5)-(4.6), which yield the claimed epsilon^(s/(s-1)) scaling (P3); the exponent is the output of the construction, not an input. Proposition 4.1 then converts (P3) to amorphic complexity via Eq. 2.5, and non-block-code-equivalence follows from the elementary Lipschitz invariance (1.3). Citations to earlier work by the present authors ([10], [19], [22], [24]) supply background facts such as Mirsky measures, taut-window aperiodicity, and the discrete spectrum of weak model sets; these are external published results with stated assumptions, and they do not enter the derivation of the new covering-number scalings as fitted parameters or as assumed conclusions. The in-text issue at Eq. (4.2) (the coding equivalence holds on the appropriate half-interval I^1_n or I^0_n rather than all of I_n) and the unstated geometric constants in (O1) are correctness and verification risks, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no invented physical entities. The parameter s in the golden-rotation family is a construction parameter, not a fitted constant; the window W_s is an explicit Borel set with a falsifiable scaling property, so it is not an ad hoc entity. All arguments rely on standard tools from ergodic theory, number theory, and harmonic analysis, listed above.

assumptions (6)
  • standard math Chinese Remainder Theorem and inclusion-exclusion for densities of arithmetic progressions, used in Lemmas 3.5-3.8.
    Used to compute densities of MB∪(r+MB) exactly.
  • standard math Davenport-Erdos theorem on densities of sets of multiples, used in Section 5.1 and Theorem 5.2.
    Ensures existence of densities for Besicovitch sets and the limit formula in Lemma 3.8.
  • domain assumption Halmos-von Neumann theorem: ergodic discrete-spectrum systems are isomorphic to rotations on compact abelian groups.
    Used in Section 1.2 and Theorem 2.9 to identify the orbit closure under the Besicovitch metric with the dual group H.
  • domain assumption Wiener-Wintner theorem and Lenz-Spindeler-Strungaru results [27] on Besicovitch almost periodicity and pure point spectrum.
    Theorem 2.9 combines these; the paper provides a self-contained proof but leans on the classical characterization of discrete spectrum.
  • domain assumption Keller-Richard [24, Thm B1']: coding of Haar aperiodic windows has pure point spectrum {e^{2πiℓα}}.
    Used in Proposition 4.1a to assert all μ_s have identical spectrum.
  • domain assumption Regular Toeplitz B-free shifts are minimal, uniquely ergodic, isomorphic extensions of their MEF with trivial automorphism group, from [10,7,19,8].
    Used in Example 3.17 to conclude the Toeplitz examples are pairwise non-isomorphic based on different amorphic complexities.

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Pith. "Pith review of Besicovitch covering numbers for $\mathcal B$-free and other shifts." pith.science (2026). https://pith.science/paper/NFQJJTZN

@misc{pith2026250509253,
  author       = {Pith},
  title        = {Pith review of: Besicovitch covering numbers for $\mathcal B$-free and other shifts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NFQJJTZN}},
  note         = {Machine review of arXiv:2505.09253}
}
abstract

For a finite alphabet $A$ define by $d_1(x,y):=\limsup_{n\to\infty}\frac{1}{2n+1}\#\{|i|\le n: x_i\neq y_i\}$ the Besicovitch pseudo-metric on $A^{\mathbb Z}$. It is well known that a closed subshift of $A^{\mathbb Z}$ has finite covering numbers w.r.t. $d_1$ if and only if it is mean-equicontinuous. Here we study, more generally, the scaling behavior of these covering numbers for individual orbits which are generic for an ergodic measure $\mu$ on $A^{\mathbb Z}$ with discrete spectrum, and we explore their usefulness as invariants for block code equivalence. We illustrate this by developing tools to determine these covering numbers for various classes of $\mathcal B$-free numbers (in particular also for square-free numbers), and we provide a continuous family of measures $\mu_s$, all with the same discrete spectrum generated by a single number, but such that $\mu_s$- and $\mu_{s'}$-typical $x$ resp. $x'\in A^{\mathbb Z}$ have sufficiently different growth of covering numbers such that there are no finite block codes mapping $x\to x'$ and $x'\to x$. (Indeed, both orbits have different amorphic complexities.)

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