REVIEW 2 major objections 4 minor 1 cited by
The coincidence of R\'{e}nyi-Parry measures for $\beta$-transformation
T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read 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 $\beta_2 = \beta_1 + 1$.
desk verdict Good paper with a real result; Theorem 1.1 needs an 'after possibly interchanging' clause before it is true. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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).
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Theorem 1.1 and Abstract] 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 2, Eq. (2.4)] 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.
minor comments (4)
- [Corollary 1.3] 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 2, after Eq. (2.5)] 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.
- [Proposition 2.6] 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.
- [Proof of Proposition 2.4] There is a typographical error: 'Lebeague' should be 'Lebesgue' in two places in the proof.
Circularity Check
No significant circularity: the main theorem is derived self-contained from the Parry density formula and elementary orbit analysis.
full rationale
The paper's central claim (Theorem 1.1) is not circular. The sufficiency direction is an explicit direct calculation (Proposition 2.1) starting from the defining density formula (2.1), and the necessity direction derives equality of normalized densities, then equality of normalization constants, then equality of initial densities (Proposition 2.4), then finiteness and equality of the nonzero orbits (Propositions 2.5 and 2.6), and finally a coefficient comparison leading to the quadratic characterization. No parameter is fitted to the desired conclusion, no result is imported from the authors' own prior work, and the cited Hochman-Shmerkin theorem is used only for Corollary 1.2 and not for the main characterization. The only notable issue is the asymmetry in the literal statement of Theorem 1.1, which omits an 'after possibly interchanging β1 and β2' clause; the proof itself inserts this as 'Without loss of generality, we may assume that 0 ∈ Oβ1 and 0 /∈ Oβ2.' That is a mathematical correctness/statement issue, not an instance of circular reasoning, because the omitted relabeling is not an input being smuggled into the conclusion. Accordingly, no circular step can be exhibited with a quote-reduction, and the appropriate score is 0.
Assumptions & free parameters
assumptions (2)
- standard math Parry's explicit density formula: the Rényi-Parry measure νβ has Radon-Nikodym derivative hβ/Kβ with hβ defined by (1.1).
- domain assumption The standard definitions of the β-transformation Tβ(x) = βx mod 1 and its orbit Oβ = {T^nβ(1) : n ≥ 1} are the objects being studied.
Cite this review
Pith. "Pith review of The coincidence of R\'{e}nyi-Parry measures for $\beta$-transformation." pith.science (2026). https://pith.science/paper/G32Z4OVI
@misc{pith2026250108116,
author = {Pith},
title = {Pith review of: The coincidence of R\'enyi-Parry measures for $\beta$-transformation},
year = {2026},
howpublished = {\url{https://pith.science/paper/G32Z4OVI}},
note = {Machine review of arXiv:2501.08116}
}
abstract
We present a complete characterization of two different non-integers with the same R\'{e}nyi-Parry measure. We prove that for two non-integers $\beta_1 ,\beta_2 >1$, the R\'{e}nyi-Parry measures coincide if and only if $\beta_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $\beta_2 = \beta_1 + 1$, which confirms a conjecture of Bertrand-Mathis in \cite[Section III]{Bertrand-1998}.
Figures
Forward citations
Cited by 1 Pith paper
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Reference graph
Works this paper leans on
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Reviewed August 10, 2026 · model on record in the stance chip above.
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