{"id":"ee11d9f4-6fda-4089-a1a6-c0f72ec35431","arxiv_id":"2507.10982","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Beatty multiple shifts are defined and their Hausdorff and Minkowski dimensions are computed explicitly for five of six parameter regions, leaving one region open.","lead":"This paper introduces a new family of symbolic dynamical systems, Beatty multiple shifts, and proves formulas for their Hausdorff and Minkowski dimensions. The result generalizes earlier dimension formulas for multiplicative and affine shifts and links the coefficients to covering properties of Beatty sequences in number theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.5(1) is unproved for α>1 and appears false for complementary Beatty parameters: α=φ, γ=φ+1 gives d2=1/φ, d∞=0, not d∞=1/φ.","rationale":"The reader's weakest assumption was the unproved orbit decomposition (2.1). I believe that decomposition is in fact valid: because f is injective and grows by factor γ/α>1, every integer either lies in A1, in a finite forward orbit of a root in S(α,β)\\S(γ,δ), or in a zero-density exceptional set. The real soft spot is sharper: the proof of Theorem 1.5(1) in Section 2 explicitly assumes α=1, yet the theorem claims d=(0,0,0,1−1/γ) for all ⟨1⟩. The complementary Beatty case with α=φ, γ=φ+1 satisfies the stated condition n/α+m/γ=1 (n=m=1) and gives, by the paper's own definitions, d2=1/φ and d∞=0. If the asserted d∞=1/φ is used in Theorem 1.4 for the full shift, the dimension exceeds 1, contradicting that X is the full shift. Thus the paper's region classification or Theorem 1.5(1) must be corrected. A direct verification of these densities settles the issue. The reader's conditional verdict is appropriate; my concern reinforces it rather than changing it.","tokens_in":15384,"tokens_out":42500,"duration_ms":444657,"concrete_test":"Verify the densities for (α,β,γ,δ)=(φ,0,φ+1,0): by Beatty's theorem, S(φ,0) and S(φ+1,0) partition N, so A2=S(φ,0), A∞=∅; hence d2=1/φ and d∞=0. If Theorem 1.5(1) gives d∞=1−1/(φ+1)=1/φ, the theorem is refuted; the authors must either correct the region assignment or explicitly exclude this boundary case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.5(1) in Section 2 explicitly assumes α=1 (it writes ⌊k+β⌋ and concludes d∞=1−1/γ), but the theorem statement covers all (α,β,γ,δ)∈⟨1⟩ as defined by condition (i) n/α+m/γ=1 with nβ/α+mδ/γ∈Z. This includes complementary Beatty pairs with α>1, e.g., α=φ, γ=φ+1, β=δ=0 (n=m=1). For this parameter set, Beatty's theorem gives S(φ,0) and S(φ+1,0) partitioning N, so S(α,β)∩S(γ,δ)=∅. By the definitions preceding Theorem 1.4, every x∈S(φ,0) lies in A2 (f(x)∈S(γ,δ)\\S(α,β)) and A∞=∅, hence d2=1/φ and d∞=0. Theorem 1.5(1) instead asserts d∞=1−1/(φ+1)=1/φ and d2=0. Substituting the asserted d into Theorem 1.4 for the full shift A (all entries 1, so X is the full shift and dim=1) yields dim=φ>1, an impossibility. Thus either the statement of Theorem 1.5(1) is false for this parameter set or the region ⟨1⟩ excludes the boundary n/α+m/γ=1 with n=m=1; the paper does not state this exclusion. This is a concrete correctness risk in the central dimension formulas, not merely an omitted proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":15801,"tokens_out":16270,"duration_ms":181294,"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":[{"comment":"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.","section":"Section 1, Theorem 1.5(1) and Figure 1"},{"comment":"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.","section":"Section 2, Eq. (2.1)"},{"comment":"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.","section":"Section 2, Proof of Theorem 1.4(2)"}],"minor_comments":[{"comment":"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.","section":"Theorem 1.4(1)"},{"comment":"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.","section":"Figure 1 and Section 1"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Section 2, Proof of Theorem 1.5(4)"}],"recommendation":"major_revision","confidential_remarks":"The counterexample in region ⟨1⟩ strongly suggests that the authors intended ⟨1⟩ to consist only of the case α=1, but the stated condition (i) in Figure 1 includes all complementary Beatty pairs with α>1. I would ask the authors to clarify the region boundaries explicitly and to verify every boundary case. I also recommend that the decomposition (2.1) be proved or else that Theorem 1.4 be restricted to a class of parameters for which it can be proved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The Beatty multiple shift is a real generalization of the earlier multiplicative and affine shifts, and the paper does well to connect its dimension theory to the classical Beatty covering problem. The Minkowski dimension counting in Theorem 1.4 is mostly self-contained, and the density computations for regions ⟨3⟩, ⟨4⟩, and ⟨5⟩ look genuinely new and likely correct. That part is worth reading carefully.\n\nThe soft spots are serious. The biggest issue is Theorem 1.5(1). The proof explicitly assumes α = 1, but the theorem statement claims all (α,β,γ,δ) in ⟨1⟩, which by condition (i) includes complementary Beatty pairs with α > 1. Take α = φ, γ = φ+1, β = δ = 0. Beatty's theorem gives S(φ,0) and S(φ+1,0) as a disjoint cover of N, so A∞ is empty and every element of S(φ,0) lands in A2. The actual density vector is d = (0, 1/φ, 0, 0), not (0, 0, 0, 1/φ). For the full shift this leads to a dimension greater than 1, impossible. So either the statement should be restricted to α = 1, or the complementary Beatty case needs a different formula. The paper does neither.\n\nSecond, the orbit decomposition (2.1) is asserted with only “it can be verified.” That is the structural backbone of the counting argument, and for real parameters it deserves a proof or a reference. The reader's conditional verdict flagged this, and I agree it is a real gap, though secondary to the Theorem 1.5(1) issue.\n\nThird, the Hausdorff dimension proof is deferred to the authors' previous paper [3]. That is acceptable if the adaptation is routine, but with a real exponent γ/α in the t-vector, existence and uniqueness of that vector are assumed rather than shown. For primitive A and ρ = γ/α > 1, the Perron–Frobenius route works, but it should be stated.\n\nFor a specialist in multiplicative subshifts, this paper contains useful ideas and a new class worth studying. But as it stands, the central dimension formulas are not fully supported, and Theorem 1.5(1) is demonstrably wrong on part of its stated domain. A serious editor should send it to peer review, but with a clear instruction to fix or restrict the ⟨1⟩ region and to prove the orbit decomposition. If the authors can do that, the paper could become a solid contribution.","headline":"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.","tokens_in":16259,"tokens_out":8105,"would_cite":false,"duration_ms":79979,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B10","37C45","28A80","11B83"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Beatty multiple shift","Hausdorff dimension","Minkowski dimension","multiplicative shift of finite type","affine multiple shift","Beatty sequence","disjoint cover","density vector"],"falsifier":"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.","tokens_in":15182,"feed_emoji":"📐","tokens_out":8743,"duration_ms":90914,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact dimensions found for Beatty multiple shifts","feed_subtitle":"Real-valued scaling parameters are covered, and the formulas recover known integer cases as a special case.","key_machinery":"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 $t_i^{\\gamma/\\alpha}=\\sum_j A(i,j)t_j$.","core_discovery":"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