{"id":"cb480640-7aa5-423e-8d9a-667956338d93","arxiv_id":"2501.08116","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For non-integer β1, β2 > 1, the Rényi-Parry measures coincide if and only if β1 is a root of x^2 − qx − p = 0 with p ≤ q and β2 = β1 + 1.","lead":"Two different non-integer number-folding maps can share the same invariant spreading pattern in exactly one family of cases: the smaller base solves x^2 = qx + p for integers p ≤ q, and the larger base is exactly one more than the smaller. The paper proves this complete classification, settling a 1998 conjecture by Bertrand-Mathis.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 as stated is not symmetric under swapping β1 and β2: the pair (α^2, α) is a counterexample to the literal statement, so the theorem needs an 'after possibly interchanging' clause.","rationale":"The reader's verdict accepts Theorem 1.1 as stated and locates the delicate part in the coefficient comparison. In good faith, I checked that coefficient-comparison step: equality of the step functions gives equality of the coefficient multisets at the common jump points, and the subsequent max/min arguments can be filled by comparing sizes of the sets, so that concern is not fatal. However, the theorem statement itself has an independent and concrete flaw: it is not invariant under exchanging β1 and β2, although the proof explicitly uses such an exchange. The counterexample (β1,β2)=(α^2,α) shows that the literal only-if direction fails, because β1 is a quadratic root of the required form but β2 is not β1+1. This does not invalidate the substantive classification of unordered pairs, which is correct and well proved after a relabeling. But a theorem stated as a universal iff over ordered pairs must include the symmetry clause. Therefore I would not reject the paper; I would accept it conditionally on correcting the statement, e.g. by adding 'after possibly interchanging β1 and β2' or by stating the characterization for unordered pairs. The reader's assumption about the weakest point is reasonable but does not address this statement-level issue, hence agreement_with_reader is 'disagree'.","tokens_in":7915,"tokens_out":21673,"duration_ms":202763,"concrete_test":"Take α=(1+√5)/2 and set (β1,β2)=(α^2,α). Verify from Proposition 2.1 that ν_{α^2}=ν_α (apply the proposition with β1=α, β2=α+1=α^2). Then check the stated condition of Theorem 1.1: β1=α^2 is a root of x^2−3x+1, so the quadratic-root part holds with p=1, q=3, but β2=α≠α^2+1=β1+1. This directly falsifies the literal 'if and only if' statement. To confirm the intended theorem, rerun the necessity proof with the swapped labels and observe that the WLOG step forces a relabeling; the theorem should be amended to include 'after possibly interchanging β1 and β2'.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central classification is mathematically correct only up to relabeling, but Theorem 1.1 omits that relabeling and is therefore false as written. Let α=(1+√5)/2, so α is a root of x^2−x−1 and α^2=α+1. Proposition 2.1 gives ν_α=ν_{α^2}. Now take the ordered pair (β1,β2)=(α^2,α). Then ν_{β1}=ν_{β2}, and β1 is a root of x^2−3x+1, which is of the form x^2−qx−p with p=1≤q=3. But β2=α is not equal to β1+1=α^2+1=α+2. Thus the theorem's 'only if' implication fails for this labeling. The proof itself contains the needed qualifier: after the hypothesis it says 'Without loss of generality, we may assume that 0∈O_{β1} and 0∉O_{β2}', which amounts to choosing the smaller element as β1. This assumption is never stated in Theorem 1.1 or the abstract. The correct statement should be that, after possibly exchanging β1 and β2, one of them is a root of x^2−qx−p with p≤q and the other equals it plus 1. This is a genuine internal flaw in the theorem statement, distinct from the fillable terseness in the coefficient-comparison step.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper characterizes the pairs of distinct non-integer bases β1, β2 > 1 for which the Rényi-Parry measures νβ1 and νβ2 coincide. The main result (Theorem 1.1) asserts that this happens exactly when β1 is a root of x^2 − qx − p with p, q ∈ N, p ≤ q, and β2 = β1 + 1. The proof uses Parry's explicit density formula, shows that equality of normalized densities forces equality of the unnormalized initial densities everywhere, then proves that the orbits of 1 must be finite with identical nonzero parts and that 0 belongs to exactly one of the two orbits. A coefficient comparison for the resulting step functions reduces the classification to the quadratic case, and the paper draws corollaries about simultaneous invariant measures (via a theorem of Hochman and Shmerkin) and about multiplicative independence.","tokens_in":8186,"tokens_out":19878,"duration_ms":171438,"significance":"If the missing relabeling clause is added, the paper gives a complete and elegant classification that confirms a conjecture of Bertrand-Mathis. The proof is elementary and self-contained, resting on the Parry density formula and a careful orbit analysis rather than on external measure-rigidity results; the Hochman-Shmerkin theorem is used only in the corollaries. The characterization is sharp and falsifiable: coincident Rényi-Parry measures for distinct non-integer bases can occur only for pairs of quadratic Pisot numbers of the described form. The main technical novelty is the reduction of the measure-coincidence problem to a discrete coefficient-comparison problem, and the main proof is sound in outline despite one terse step.","major_comments":[{"comment":"The statement is false as written because it omits a relabeling. Let α = (1+√5)/2. By Proposition 2.1 applied to β1 = α (root of x^2 − x − 1, p = q = 1) and β2 = α + 1 = α^2, we have ν_α = ν_{α^2}. Thus the ordered pair (β1, β2) = (α^2, α) satisfies ν_{β1} = ν_{β2}. Here β1 = α^2 is a root of x^2 − 3x + 1 (p = 1 ≤ q = 3), but β2 = α is not equal to β1 + 1 = α^2 + 1. Hence the 'only if' direction fails for this labeling. The proof itself begins the necessity argument with 'Without loss of generality, we may assume that 0 ∈ O_{β1} and 0 ∉ O_{β2}', which is precisely an exchange of the two bases. The theorem and the abstract must state that the characterization holds after possibly interchanging β1 and β2.","section":"Theorem 1.1 and Abstract"},{"comment":"The passage from equality of the step functions in (2.2) and (2.3) to equality of the coefficient sets in (2.4) is too terse. Since hβ1 = hβ2 everywhere, the right-continuous step representations have the same jump points and the same jump size at each point, so the multiset of coefficients in (2.2) equals the multiset formed by the coefficients in (2.3). Because the values {1/β1^k : 1 ≤ k ≤ m} are strictly decreasing and hence distinct, this multiset equality implies the set equality C = C1 ∪ C2. This justification should be supplied, as the subsequent max/min argument relies on it.","section":"Section 2, Eq. (2.4)"}],"minor_comments":[{"comment":"The proof invokes Theorem 1.1 for the given ordered pair (β1, β2). Once Theorem 1.1 is corrected to allow an exchange of the two bases, the argument should first note that the desired conclusion is symmetric and then apply the theorem to the properly ordered pair.","section":"Corollary 1.3"},{"comment":"The deduction β1 < β2 from hβ1(0) = hβ2(0) uses the fact that x ↦ x/(x − 1) is strictly decreasing on (1, ∞); this monotonicity should be stated explicitly.","section":"Section 2, after Eq. (2.5)"},{"comment":"When ruling out the case 0 ∈ Oβ1 ∩ Oβ2, the proof should explicitly invoke the strict monotonicity of the partial geometric sums ∑_{k=0}^{n} β^{-k} in β to conclude β1 = β2 from equality of the sums.","section":"Proposition 2.6"},{"comment":"There is a typographical error: 'Lebeague' should be 'Lebesgue' in two places in the proof.","section":"Proof of Proposition 2.4"}],"recommendation":"major_revision","confidential_remarks":"The main mathematical content is sound and the proof is essentially correct, but the central theorem statement is false without the relabeling clause. This is an easily fixable statement-level error rather than a defect in the proof strategy, so I recommend major revision rather than rejection. The authors should also expand the coefficient-comparison step in the proof of Theorem 1.1, since it is currently too compressed for the burden it carries."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper's classification is correct and the proof is largely self-contained, but Theorem 1.1 as stated is missing an 'after possibly interchanging β1 and β2' clause, so it is literally false. That is a one-line fix and doesn't undermine the substance, but it has to be made before this is citable.\n\nThe genuinely new thing is the necessity direction: starting from equality of the Rényi-Parry densities, they show the orbits of 1 are finite, the nonzero parts coincide, and 0 belongs to exactly one orbit. That forces the quadratic relation and β2 = β1 + 1. I checked the sufficiency calculation and the finite-orbit arguments; they are correct. The normalization constants are handled properly via right-continuity. The paper confirms Bertrand-Mathis's conjecture and gives a clean application to the Hochman–Shmerkin rigidity question. No fitting, no circularity; the argument stands on the classical Parry density formula.\n\nThe soft spots: first, the symmetry issue. Let α be the golden ratio. Proposition 2.1 gives ν_α = ν_{α^2}. Now take (β1, β2) = (α^2, α). The theorem's conclusion fails: α^2 is not the root of any x^2 − qx − p with p, q ∈ N (its minimal polynomial is x^2 − 3x + 1, which would require p = −1). The proof itself makes the WLOG choice '0 ∈ O_{β1}', which selects the smaller element, so the intended statement is correct up to relabeling. That should be in the theorem and the abstract. (The stress-test note gets the conclusion right but the algebra in its explanation is off.)\n\nSecond, the coefficient comparison around (2.4) is terser than it should be. The step functions agree as right-continuous functions, so the coefficient at each common jump point must match; 'sets of coefficients' is a bit sloppy and should be 'coefficients at each ordered jump point'. This is fillable, but a referee will want it spelled out.\n\nMinor: p, q should be explicitly positive integers.\n\nBottom line: this deserves serious peer review. It's a real result, readable by anyone in dynamical systems, and the flaws are presentation-level. With the relabeling clause and a slightly expanded coefficient-matching argument, I'd accept.","headline":"Good paper with a real result; Theorem 1.1 needs an 'after possibly interchanging' clause before it is true.","tokens_in":8720,"tokens_out":7998,"would_cite":true,"duration_ms":72099,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For two different non-integer parameters β1, β2 > 1, the Rényi-Parry measures of the β-transformations coincide precisely when β1 is a root of $x^2 - qx - p = 0$ with positive integers $p \\le q$ and $\\beta_2 = \\beta_1 + 1$.","keywords":["beta-transformation","Rényi-Parry measure","Pisot number","invariant measure","density function","beta-expansion"],"falsifier":"A single counterexample would settle the claim: exhibit two distinct non-integers $\\beta_1, \\beta_2 > 1$ whose Rényi-Parry densities agree at every point but for which $\\beta_1$ is not a root of $x^2 - qx - p = 0$ with $p, q \\in \\mathbb{N}$ and $p \\le q$, or for which $\\beta_2 \\neq \\beta_1 + 1$. Concretely, evaluating $h_{\\beta_1}$ and $h_{\\beta_2}$ at the jump points of their common orbit would reveal any coefficient mismatch.","tokens_in":7688,"feed_emoji":"📐","tokens_out":5948,"duration_ms":56646,"temperature":0.7,"pith_summary":"The paper settles when two different non-integer $\\beta$-transformations can have the same invariant absolutely continuous measure, the Rényi-Parry measure. The answer is that this happens almost never, and when it does, the two parameters are forced into a tight algebraic relationship: the smaller solves a quadratic equation $x^2 - qx - p = 0$ with $p, q \\in \\mathbb{N}$, $p \\le q$, and the larger is exactly one more. The proof is carried out by reading the density function of the measure from the orbit of the point 1 and comparing the coefficients attached to the jump points of these step functions. This complete characterization confirms a conjecture stated in 1998 and closes a question left open in the rigidity theory of $\\beta$-transformations.","feed_headline":"Only shifted quadratic roots share beta-map measures","feed_subtitle":"A complete answer: non-integer beta maps coincide only for a quadratic root and its successor by one.","key_machinery":"The central object is the initial density function $h_\\beta(x) = \\sum_{x < T_\\beta^n(1)} \\beta^{-n}$ on $[0,1)$, before normalization by the constant $K_\\beta$. This function is a decreasing, right-continuous step function whose jump points are exactly the orbit points $T_\\beta^n(1)$, and its normalized version is the Radon--Nikodym derivative of the Rényi-Parry measure. The proof converts equality of the measures into pointwise equality of these densities, then into equality of the orbit sets and of the coefficient sets in the step-function representation. The algebraic conclusion comes from comparing the largest and smallest coefficients in the sets displayed as equation (2.4).","core_discovery":"Theorem 1.1 states that for two distinct non-integers $\\beta_1, \\beta_2 > 1$, the Rényi-Parry measures $\\nu_{\\beta_1}$ and $\\nu_{\\beta_2}$ coincide if and only if $\\beta_1$ is the root of $x^2 - qx - p = 0$ for some $p, q \\in \\mathbb{N}$ with $p \\le q$, and $\\beta_2 = \\beta_1 + 1$. In particular, whenever the measures coincide, both parameters are Pisot numbers of degree 2. The necessity proof shows that equality of the measures forces the orbits of 1 under the two transformations to agree except for whether 0 belongs to the orbit, then forces equality of the coefficient sets in the density step functions, and finally uses a maximum--minimum analysis of those coefficients to derive the quadratic equation and the shifted relation.","pith_inferences":["The coefficient-set equality (2.4) suggests a broader rigidity: if two non-integer beta-maps had proportional, rather than equal, Rényi-Parry densities, a similar comparison of jump coefficients might force an algebraic relation of the same quadratic type; the paper does not address this.","The theorem may extend to a statement about beta-shifts: because the Rényi-Parry measure is the unique maximal-entropy measure, coincident measures could imply strong isomorphism properties of the corresponding symbolic systems under the same quadratic condition, though this is not explored here.","A testable consequence for numerical experiments is that any pair of distinct non-integers not of the stated quadratic form should always show a mismatch between $h_{\\beta_1}$ and $h_{\\beta_2}$ at some point in the common candidate orbit, so direct evaluation of these step functions can quickly rule out measure coincidence."],"forward_implications":["For non-integer beta-maps, coincident Rényi-Parry measures occur only in the countable family of pairs $(\\beta, \\beta+1)$ where $\\beta$ is a quadratic Pisot root of $x^2 - qx - p$ with $p, q \\in \\mathbb{N}$ and $p \\le q$.","If $\\beta_1$ is a Pisot number of degree at least 3 and is multiplicatively independent from $\\beta_2$, then no Borel probability measure can be jointly invariant under $T_{\\beta_1}$ and $T_{\\beta_2}$ and ergodic with positive entropy under $T_{\\beta_2}$, as stated in Corollary 1.2.","If $\\nu_{\\beta_1} = \\nu_{\\beta_2}$ and $\\log \\beta_1 / \\log \\beta_2$ is rational, the only possible pair is the golden ratio and its square, as stated in Corollary 1.3.","The proof gives a direct criterion for equality of the invariant measures that does not rely on ergodic-theoretic rigidity: the unnormalized density step functions themselves must be equal.","Since the Rényi-Parry measure is the unique measure of maximal entropy for each beta-map, the characterization also identifies exactly when two non-integer beta-shifts have the same measure of maximal entropy."],"supporting_citations":[{"why":"States the conjecture, in its Section III, that this paper confirms, thereby supplying the problem and the expected characterization.","marker":"[1]"},{"why":"Provides the rigidity theorem for Pisot parameters and raises the question of exactly which pairs have coincident Rényi-Parry measures.","marker":"[2]"},{"why":"Gives the explicit formula for the density function $h_\\beta$ that is the paper's main working tool.","marker":"[4]"},{"why":"Introduces $\\beta$-transformations and proves the existence and uniqueness of the invariant measure $\\nu_\\beta$ equivalent to Lebesgue measure.","marker":"[5]"}],"fun_headline_variants":["Beta measures match only when beta2 = beta1+1","Shared beta measures imply quadratic root shifted by one","Coincident Rényi-Parry: exactly root and root+1","Two beta-transformations merge only for shifted quadratic roots","Beta map coincidence classified: quadratic shift pairs only"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The necessity proof assumes that two equal step functions built from the same set of jump points must have matching coefficients at each point, so that the largest coefficient can be identified by value alone.","fun_headline_variants_meta":{"raw":{"variants":["Beta measures match only when beta2 = beta1+1","Shared beta measures imply quadratic root shifted by one","Coincident Rényi-Parry: exactly root and root+1","Two beta-transformations merge only for shifted quadratic roots","Beta map coincidence classified: quadratic shift pairs only"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000874,"raw_usage":{"total_tokens":3727,"prompt_tokens":838,"completion_tokens":2889,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":2808}},"tokens_in":454,"tokens_out":2889,"duration_ms":20517,"temperature":1.0,"reasoning_tokens":2808,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:31:01.115108+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A single counterexample would settle the claim: exhibit two distinct non-integers $\\beta_1, \\beta_2 > 1$ whose Rényi-Parry densities agree at every point but for which $\\beta_1$ is not a root of $x^2 - qx - p = 0$ with $p, q \\in \\mathbb{N}$ and $p \\le q$, or for which $\\beta_2 \\neq \\beta_1 + 1$. Concretely, evaluating $h_{\\beta_1}$ and $h_{\\beta_2}$ at the jump points of their common orbit would reveal any coefficient mismatch.","supporting_citations":[{"cited_title":"Bertrand-Mathis","cited_arxiv_id":null,"evidence_quote":"States the conjecture, in its Section III, that this paper confirms, thereby supplying the problem and the expected characterization."},{"cited_title":"Hochman and P","cited_arxiv_id":null,"evidence_quote":"Provides the rigidity theorem for Pisot parameters and raises the question of exactly which pairs have coincident Rényi-Parry measures."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces $\\beta$-transformations and proves the existence and uniqueness of the invariant measure $\\nu_\\beta$ equivalent to Lebesgue measure."}],"review_version":1}