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Hausdorff dimensions of Beatty multiple shifts

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Pith's one-line read The paper introduces Beatty multiple shifts—subshifts constrained along two Beatty sequences—and proves explicit Hausdorff and Minkowski dimension formulas that depend only on the transition matrix and a density vector of the integer…

desk verdict New object and mostly plausible formulas, but Theorem 1.5(1) appears false for complementary Beatty pairs; the paper needs revision before it can be trusted. read the letter →

arxiv 2507.10982 v1 pith:55IPIGIC submitted 2025-07-15 math.DS math.NT

Jung-Chao Ban , Wen-Guei Hu , Guan-Yu Lai This is my paper · ORCID
classification math.DSmath.NT MSC 37B1037C4528A8011B83
keywords BeattymultipleshiftHausdorffdimensionMinkowskimultiplicativeoffinitetypeaffinesequencedisjointcoverdensityvector
topics The Collatz Conjecture
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 introduces the Beatty multiple shift, a set of infinite symbol sequences in which allowed transitions are checked at positions given by two Beatty sequences, $\lfloor\alpha k+\beta\rfloor$ and $\lfloor\gamma k+\delta\rfloor$, with $1\le\alpha<\gamma$. It proves that, whenever the associated density vector $d=(d_1,d_2,\ldots,d_\infty)$ exists, the Minkowski dimension has a closed formula involving the sums of entries of powers of the transition matrix, and that for primitive matrices the Hausdorff dimension has a formula involving a positive vector $t$ satisfying $t_i^{\gamma/\alpha}=\sum_j A(i,j)t_j$. The formulas recover the earlier multiple shift of finite type and affine multiple shift results as integer-parameter cases, and they show that the two dimensions coincide exactly when the row sums of $A$ are equal. The paper also computes the density vector in all but one parameter region, leaving region $\langle6\rangle$ as an explicit open problem.

What carries the argument

The machine is the orbit decomposition of the positive integers induced by the map $f$ defined on $S(\alpha,\beta)=\{\lfloor\alpha k+\beta\rfloor:k\in\mathbb{N}\}$ by $f(\lfloor\alpha k+\beta\rfloor)=\lfloor\gamma k+\delta\rfloor$. The sets $A_i$ collect integers whose $f$-orbit passes through the overlap $S(\alpha,\beta)\cap S(\gamma,\delta)$ for exactly $i-1$ steps before landing outside, and $A_\infty$ collects integers with infinite orbits; their densities $d_i$ and $d_\infty$ are the only input beyond $A$. Each orbit segment $\{x,f(x),\ldots,f^{\ell}(x)\}$ imposes $|A^\ell|$ possible symbol choices, so the cylinder-counting product factors over orbit segments and yields the Minkowski formula. The Hausdorff argument transplants a Markov measure construction: the measure is built on each orbit segment from the same matrix products and from the normalizing constants $t_{\emptyset,i}$, and on infinite orbits from the positive vector $t$ fixed by the power map $t_i^{\gamma/\alpha}=\sum_j A(i,j)t_j$.

What would settle it

Take a concrete unresolved tuple, for instance $\alpha=\sqrt{2}$, $\gamma=\sqrt{3}$ with $\beta=0$ and $\delta=1/2$, and a small primitive matrix $A$; estimate the box-counting slope of admissible words up to length $n$ by direct enumeration and compare it with the value predicted by Theorem 1.4 using densities computed from the orbit decomposition. A mismatch, or a direct computation showing that the remainder $R$ in (2.1) has positive density, would settle whether the formula is correct.

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Extended reading notes

Core claim

The central claim is that the fractal dimension of $X_A^{[\alpha,\beta,\gamma,\delta]}$ is completely determined by the transition matrix $A$ and by how the positive integers split into forward orbits of the map $f(\lfloor\alpha k+\beta\rfloor)=\lfloor\gamma k+\delta\rfloor$. Writing $d_i$ for the density of integers whose orbit under $f$ has exactly $i$ points before leaving a specified region, and $d_\infty$ for the density of infinite orbits, the Minkowski dimension equals $$\dim_M $X_A^{{[\alpha,\beta,\gamma,\delta]}}$ = \sum_{i=1}^{\infty} \left[ \frac{1}{(\gamma/\$\alpha$)^{i-1}} d_i + \left(\frac{1}{(\gamma/\$\alpha$)^{i-1}} - \frac{1}{(\gamma/\$\alpha$)^i}\right)\left(\sum_{j>i} d_j + d_\infty\right)\right] \log_m |$A^{{i-1}}$|,$$ for irreducible $A$. For primitive $A$, the Hausdorff dimension equals $$\dim_H $X_A^{{[\alpha,\beta,\gamma,\delta]}}$ = d_1 + \sum_{i=2}^{\infty} d_i \log_m t_{\emptyset,i} + d_\infty \log_m \sum_{i=0}^{m-1} t_i,$$ where $t$ is the unique positive vector with $t_i^{\gamma/\alpha}=\sum_j A(i,j)t_j$. Equality of the two dimensions holds if and only if the row sums of $A$ are all equal. This is an extension: the previously treated cases with integer parameters $(p,a,q,b)$ and $p<q$ are special choices of the real parameters.

Load-bearing premise

The proof assumes that the positive integers are, up to a zero-density set, the disjoint union of the finite and infinite orbits of $f$ used in (2.1); the paper asserts this 'can be verified' but supplies no proof, and the counting argument collapses if the decomposition has a positive-density remainder.

Editorial extensions

If this is right

  • For integer parameters $(\alpha,\beta,\gamma,\delta)=(p,a,q,b)$, Theorem 1.4 collapses to the known dimension formulas for affine multiple shifts and, when $p=1$ and $a=b=0$, to the original multiple shift of finite type.
  • Because the densities $d_i$ are computed in regions $\langle1\rangle$ through $\langle5\rangle$ and $\langle7\rangle$ through $\langle10\rangle$, the dimension formulas become fully explicit for all those parameter choices.
  • The equality criterion $\dim_H X_A^{[\alpha,\beta,\gamma,\delta]}=\dim_M X_A^{[\alpha,\beta,\gamma,\delta]}$ holds exactly when the rows of $A$ sum to a common value, matching the classical multiple-SFT situation.
  • The unresolved region $\langle6\rangle$ is the sole obstacle to a complete classification; Problem 1 states that determining the density vector $d$ there is open.

Reading between the lines

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

  • If the orbit decomposition can be proved for all real parameter choices, the dimension formulas would hold for every tuple with $1\le\alpha<\gamma$; the currently unproved step is the claim that the remainder $R$ in (2.1) has zero density.
  • The unresolved region $\langle6\rangle$ likely requires finer Diophantine information about $\alpha$ and $\gamma$ than uniform distribution alone, since the resolved regions are exactly those where the Beatty sequences behave like disjoint, aligned, or arithmetic-progression-like sets.
  • One testable extrapolation is that for any fixed $A$, two parameter tuples sharing the same density vector produce identical dimension values, so the phase shifts $\beta$ and $\delta$ matter only through their effect on $d$.
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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 / 4 minor

Summary. The paper introduces the Beatty multiple shift X_A^{[α,β,γ,δ]}, defined by the constraint A(x_{⌊αk+β⌋}, x_{⌊γk+δ⌋})=1 for 1≤α<γ, as a common generalization of the multiplicative shift of finite type of Kenyon--Peres--Solomyak and the affine multiple shift of Ban--Hu--Lai--Liao. The main results, Theorem 1.4, give Minkowski and Hausdorff dimension formulas for X_A^{[α,β,γ,δ]} in terms of densities d_i of certain orbit classes A_i and A_∞ under the map f(⌊αk+β⌋)=⌊γk+δ⌋. Theorem 1.5 then partitions the parameter space into regions ⟨1⟩,...,⟨10⟩ and computes the density vector d for all but one region, with the remaining region left as an open problem. The proofs of the dimension formulas use a decomposition of N into disjoint forward orbits of f, and the Hausdorff dimension part is imported from the authors' earlier paper [3].

Significance. If the results were correct and fully proved, the paper would give a valuable bridge between Beatty-sequence combinatorics and the dimension theory of multiplicative subshifts. The explicit nature of the formulas and the identification of the density vector d as the key object are attractive strengths, and the paper honestly identifies an open case. However, the current manuscript contains a concrete counterexample to the stated density vector in region ⟨1⟩, the central orbit decomposition is asserted without proof, and the Hausdorff dimension argument relies on an unproved extension of results from [3] to real exponents. The significance is therefore conditional on repairing these load-bearing points.

major comments (3)
  1. [Section 1, Theorem 1.5(1) and Figure 1] The asserted density vector d=(0,0,0,1−1/γ) for region ⟨1⟩ is contradicted by the complementary Beatty pair α=φ, γ=φ+1, β=δ=0. This parameter set satisfies condition (i) in Figure 1 with n=m=1, since 1/φ+1/(φ+1)=1 and 0∈Z. By Beatty's theorem, S(φ,0) and S(φ+1,0) partition N, so A1=∅, A2=S(φ,0), and A∞=∅, giving d=(0,1/φ,0,0), not (0,0,0,1/φ). Substituting the paper's vector into Theorem 1.4(1) for the full shift A with all entries 1 gives dimension φ>1, which is impossible because X_A^[α,β,γ,δ] is then the full shift of dimension 1. Either Theorem 1.5(1) is false for α>1 or region ⟨1⟩ must exclude complementary pairs with α>1; the paper states no such exclusion.
  2. [Section 2, Eq. (2.1)] The decomposition N = ⊔_{i≥1} ⊔_{x∈A_i} {x,...,f^{i-1}(x)} ⊔ ⊔_{x∈A∞} {x,f(x),...} ⊔ R is asserted with only the comment that it 'can be verified' and that R has zero density. This decomposition is the foundation of the entire Minkowski counting argument in the proof of Theorem 1.4, and the dimension formulas collapse if it fails for general real α, γ, β, δ. The paper does not prove disjointness of the orbits, the zero-density property of R, or the validity of the decomposition for irrational parameters such as the complementary pair α=φ, γ=φ+1. A complete proof of (2.1), or a precise statement of the hypotheses under which it holds, is required.
  3. [Section 2, Proof of Theorem 1.4(2)] The Hausdorff dimension proof is not self-contained: it says 'by the similar process as proof of [3, Theorem 1.3 (2)]' and defines measures µ_i and µ∞ using quantities t_{∅,i} and f_k that are introduced only inside the proof. More importantly, the theorem asserts the existence and uniqueness of a positive vector t satisfying t_i^{γ/α}=Σ_j A(i,j)t_j for the real exponent γ/α>1, but no proof is given. The results cited from [3] are for integer affine parameters, so the extension to real exponents requires a genuinely new Perron--Frobenius-type argument. Without this, the Hausdorff formula in Theorem 1.4(2) is not established.
minor comments (4)
  1. [Theorem 1.4(1)] In the displayed formula, the expression Σ_{j=i+1}^∞ d_i should presumably read Σ_{j=i+1}^∞ d_j; as printed, the sum over j of a constant d_i is a typo.
  2. [Figure 1 and Section 1] The regions ⟨1⟩,...,⟨6⟩ are described only through the cryptic caption of Figure 1; the text never gives an explicit set-theoretic definition of these regions. This makes it impossible to verify which boundary cases belong to which region, which is precisely where the counterexample in Theorem 1.5(1) arises.
  3. [Throughout] There are several typos and notational infelicities: 'Minskowski' in the introduction, unusual spacing in the title 'BEA TTY MUL TIPLE SHIFTS', and the notation '0=(0)_{i≥3}' in Theorem 1.5(1) is confusing because d is an infinite sequence.
  4. [Section 2, Proof of Theorem 1.5(4)] The proof uses constants c and d with expressions like '|A2∩[n1,n2]| = |S(α,β)∩[n1,n2]| ± 2c'; the signs and the dependence of c on the parameters are not quantified, which makes the estimate hard to check.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dimension formulas are derived from independent Beatty-sequence densities; the noted gaps are correctness risks, not construction-level circularity.

full rationale

The central derivation is not circular. Theorem 1.4 is proved by counting cylinder sets via the orbit decomposition (2.1) and the density limits (2.8), and the only external input sequence d is obtained independently from Beatty-sequence and uniform-distribution results (Theorem 1.5 and Section 3), not by fitting the dimension formula. The Minkowski half is self-contained modulo the asserted but unproved zero-density remainder R in (2.1); that is an omitted proof, not a circular step. The Hausdorff half delegates its measure construction to the authors' prior paper [3] with 'by the similar process as proof of [3, Theorem 1.3 (2)]'; this is a legitimate citation to an independent published theorem on affine multiple shifts, not a reduction of the present claim to its own inputs, and it does not make the dimension formula an input. I also weighed the manuscript's own apparent limitations: the decomposition (2.1) is asserted with only 'it can be verified', and the proof of Theorem 1.5(1) explicitly assumes alpha = 1, while the region ⟨1⟩ as drawn includes complementary Beatty pairs with alpha > 1, for which the stated densities appear incorrect. These are correctness and rigor concerns, not circularity: no equation is shown to be equivalent to its own input by definition, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper introduces no fitted parameters. Its central claims rely on the orbit decomposition (2.1), the existence and uniqueness of the positive t-vector, and cited number-theoretic results. Region ⟨6⟩ is explicitly unresolved, so the explicit dimension formulas are conditional on the density vector d being determined.

assumptions (4)
  • ad hoc to paper Decomposition (2.1): N = ⊔_i ⊔_{x∈A_i} {x,...,f^{i-1}(x)} ⊔ ⊔_{x∈A∞} {x,f(x),...} ⊔ R, with R of zero density.
    Both dimension formulas rely on this partition of the integers into forward orbits of f; the paper says 'it can be verified' and supplies only a sketch.
  • domain assumption There is a unique positive vector t satisfying t_i^{γ/α} = Σ_j A(i,j)t_j for real γ/α > 1 and primitive A.
    Stated in Theorem 1.4(2) without proof or citation; needed to define the Hausdorff dimension formula.
  • domain assumption Harman's theorems on intersections of Beatty sequences ([16, Theorems 6, 8]) imply |S(α,β) ∩ S(γ,δ)| ≤ 1 in region ⟨5⟩.
    Used in the proof of Theorem 1.5(4) to compute d_1 and d_2; accepted as an external result.
  • standard math Equidistribution of fractional parts {x/α} and {x/γ} in [0,1) and [0,1)^2, used to compute d in regions ⟨3⟩ and ⟨4⟩.
    Invoked in the proof of Theorem 1.5(3); standard result but not derived in this paper.
invented entities (1)
  • Beatty multiple shift X_A^{[α,β,γ,δ]}
    purpose: New class of subshifts whose allowed sequences are constrained on pairs of positions given by Beatty sequences; the object of all main theorems.
    It is a definition, not a posited physical entity; its properties are established by the paper's proofs, so it has no independent falsifiable handle beyond the paper itself.

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Pith. "Pith review of Hausdorff dimensions of Beatty multiple shifts." pith.science (2026). https://pith.science/paper/55IPIGIC

@misc{pith2026250710982,
  author       = {Pith},
  title        = {Pith review of: Hausdorff dimensions of Beatty multiple shifts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/55IPIGIC}},
  note         = {Machine review of arXiv:2507.10982}
}
read the original abstract

In this paper, the Beatty multiple shift is introduced, which is a generalization of the multiplicative shift of finite type (multiple SFT) [Kenyon, Peres and Solomyak, Ergodic Theory and Dynamical Systems, 2012] and the affine multiple shift [Ban, Hu, Lai and Liao, Advances in Mathematics, 2025]. The Hausdorff and Minkowski dimension formulas are obtained, and the coefficients of the formula is closely related to the classical disjoint covering of the positive integers in number theory.

Figures

Figures reproduced from arXiv: 2507.10982 by the authors.

Figure 1
Figure 1. (i) n α + m γ = 1 and nβ α + mδ γ ∈ Z, and (ii) n α − m γ = 0 with (n, m) = 1 and 1 − m α ≥ n m(β−δ) α o ≥ m α . The dashed line means the cases when it satisfies the conditions under such line. (1) If (α, β, γ, δ) ∈ ⟨1⟩, then d = (0, 0, 0, 1 − 1 γ ), where 0 = (0)i≥3. (2) If (α, β, γ, δ) ∈ ⟨2⟩, then d is defined in Section 3. (3) If (α, β, γ, δ) ∈ ⟨3⟩ ∪ ⟨4⟩, then d =  (α − 1) (γ − 1) αγ , (α − 1) (γ − 1) α2γ , d, … view at source ↗

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Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [3]

    Ban, W.-G

    J.-C. Ban, W.-G. Hu, G.-Y. Lai, and L.-M. Liao, Hausdorff dimensions of affine multiplicative subshifts, Advances in Mathematics 471 (2025). HAUSDORFF DIMENSIONS OF BEATTY MULTIPLE SHIFTS 17

    work page 2025

  2. [1]

    Ban, C.-H

    J.-C. Ban, C.-H. Chang, W.-G. Hu, G.-Y. Lai, and Y.-L. Wu, Topologically mixing properties of multiplicative integer systems. , Real Analysis Exchange 47 (2022), no. 1

    work page 2022

  3. [2]

    Ban, W.-G

    J.-C. Ban, W.-G. Hu, and G.-Y. Lai, Hausdorff dimension of multidimensional multiplicative subshifts , Ergodic Theory and Dynamical Systems 44 (2024), no. 5, 1239–1268

    work page 2024

  4. [4]

    Ban, W.-G

    J.-C. Ban, W.-G. Hu, and S.-S. Lin, Pattern generation problems arising in mul- tiplicative integer systems, Ergodic Theory and Dynamical Systems 39 (2019), no. 5, 1234–1260

    work page 2019

  5. [5]

    Brunet, Dimensions of self-affine sponges invariant under the action of mul- tiplicative integers, Ergodic Theory and Dynamical Systems (2021), 1–43

    G. Brunet, Dimensions of self-affine sponges invariant under the action of mul- tiplicative integers, Ergodic Theory and Dynamical Systems (2021), 1–43

    work page 2021

  6. [6]

    thesis, Universit´ e Paris-Nord-Paris XIII, 2022

    , Contribution to the dimensional theory of statistically self-affine car- pets and sponges or invariant under the action of multiplicative integers , Ph.D. thesis, Universit´ e Paris-Nord-Paris XIII, 2022

    work page 2022

  7. [7]

    Carinci, J

    G. Carinci, J. R. Chazottes, C. Giardina, and F. Redig, Nonconventional aver- ages along arithmetic progressions and lattice spin systems, Indagationes Math- ematicae 23 (2012), no. 3, 589–602

    work page 2012

  8. [8]

    Chatterjee and A

    S. Chatterjee and A. Dembo, Nonlinear large deviations , Advances in Mathe- matics 299 (2016), 396–450

    work page 2016

Show all 23 references
  1. [9]

    J. R. Chazottes and F. Redig, Thermodynamic formalism and large deviations for multiplication-invariant potentials on lattice spin systems , Electronic Jour- nal of Probability 19 (2014)

    2014

  2. [10]

    Fan, Some aspects of multifractal analysis , Geometry and Analysis of Fractals, Springer, 2014, pp

    A.-H. Fan, Some aspects of multifractal analysis , Geometry and Analysis of Fractals, Springer, 2014, pp. 115–145

    2014

  3. [11]

    , Multifractal analysis of weighted ergodic averages , Advances in Math- ematics 377 (2021), 107488

    2021

  4. [12]

    A.-H. Fan, J. Schmeling, and M. Wu, Multifractal analysis of multiple ergodic averages, Comptes Rendus. Math´ ematique349 (2011), no. 17-18, 961–964

    2011

  5. [13]

    1, 17–26

    , Level sets of multiple ergodic averages , Monatshefte f¨ ur Mathematik 168 (2012), no. 1, 17–26

    2012

  6. [14]

    , Multifractal analysis of some multiple ergodic averages , Advances in Mathematics 295 (2016), 271–333

    2016

  7. [15]

    R. L. Graham, Covering the positive integers by disjoint sets of the form {[nα+ β] : n = 1, 2, ...}, Journal of Combinatorial Theory, series A 15 (1973), no. 3, 354–358

    1973

  8. [16]

    Harman, Primes in intersections of beatty sequences

    G. Harman, Primes in intersections of beatty sequences. , Journal of Integer Sequences 18 (2015), no. 7, 15–7

    2015

  9. [17]

    2, 572–586

    , Primes in beatty sequences in short intervals , Mathematika 62 (2016), no. 2, 572–586

    2016

  10. [18]

    Kenyon, Y

    R. Kenyon, Y. Peres, and B. Solomyak, Hausdorff dimension for fractals in- variant under multiplicative integers , Ergodic Theory and Dynamical Systems 32 (2012), no. 5, 1567–1584

    2012

  11. [19]

    Kifer, Lectures on large deviations in probability and dynamical systems , Hyperbolic Dynamics, Fluctuations and Large Deviations 89 (2015), 45

    Y. Kifer, Lectures on large deviations in probability and dynamical systems , Hyperbolic Dynamics, Fluctuations and Large Deviations 89 (2015), 45

    2015

  12. [20]

    Kifer and S

    Y. Kifer and S. Varadhan, Nonconventional large deviations theorems , Proba- bility Theory and Related Fields 158 (2014), no. 1, 197–224

    2014

  13. [21]

    Kuipers and H

    L. Kuipers and H. Niederreiter, Uniform distribution of sequences , Courier Corporation, 2012

    2012

  14. [22]

    Peres, J

    Y. Peres, J. Schmeling, S. Seuret, and B. Solomyak,Dimensions of some fractals defined via the semigroup generated by 2 and 3 , Israel Journal of Mathematics 199 (2014), no. 2, 687–709

    2014

  15. [23]

    Peres and B

    Y. Peres and B. Solomyak, Dimension spectrum for a nonconventional ergodic average, Real Analysis Exchange 37 (2012), no. 2, 375–388. 18 JUNG-CHAO BAN, WEN-GUEI HU, AND GUAN-YU LAI (Jung-Chao Ban) Department of Mathematical Sciences, National Chengchi Univer- sity, Taipei 1160...

    2012

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