{"id":"10d98c85-19e7-48ac-9c9a-fe3a10e6ac2e","arxiv_id":"1908.09023","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For Pisot bases and sofic digit shifts, generalized Erdős measures are always pure, and absolute continuity is equivalent to vanishing of the Fourier limits along beta powers.","lead":"This paper studies measures made by pushing the standard random-digit measure on a finite automaton's output through the map that turns digit strings into real numbers in a Pisot base. It claims such measures are always pure, and that absolute continuity is equivalent to a Fourier-limit condition, though key proof steps are missing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Lemma 1 asserts without justification that the preimage of the atomic or singular support is shift-invariant; under the digital-map cocycle this requires βP−A⊆P, a condition never stated, so Theorem 1 (purity) is unproven.","rationale":"The reader's verdict is REJECT, and I agree the paper has a serious proof gap, but the specific weakest assumption named by the reader is not the right one. The claim that a singular measure on R can project to Lebesgue mod 1 via a singular function lift g(t)=t+n(t) is incorrect: any measurable integer-valued perturbation of the identity pushes Lebesgue forward to an absolutely continuous measure, because the preimage of a null set is a countable union of translated null sets. More generally, if a probability measure on R^r has Lebesgue as its mod 1 projection, the disintegration into atomic fiber measures shows it is a countable sum of translated Lebesgue pieces and is therefore absolutely continuous. Hence the step in Theorem 5 from 'ψ is Lebesgue' to 'ν is absolutely continuous' is valid, albeit unstated. The real gap is Lemma 1, used for Theorem 1. The proof's assertion that X^{-1}(D) is shift-invariant is not justified and fails for the digital-map cocycle unless β(D−A)⊆D, a condition not proved. Since Theorem 1 is a central claimed result and its proof is invalid as written, the REJECT verdict stands, though for this reason rather than the reader's Theorem 5 objection.","tokens_in":12838,"tokens_out":50723,"duration_ms":506058,"concrete_test":"Use the automaton of Example 1 (Figure 1) with β = (1+√5)/2, digits {0,±1}. Enumerate all finite paths realizing the atom at value 1 and check whether their one-step shifts σx realize a value in the atom set P = {0, ±1, ±1/β}. If any such shift lands outside P, the set (φ+)^{-1}(P) is not shift-invariant, directly refuting the assertion in the proof of Lemma 1. This check can be done by exhaustive search over paths of length up to the number of states, since the automaton is finite and P is known.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's Theorem 1 (purity) rests on Lemma 1, which claims that for any ergodic shift-invariant measure and convergent series X = Σ X_n with X_n depending on the n-th coordinate, the distribution is pure. The proof asserts that X^{-1}(D) is shift-invariant for the support D of the atomic part, and similarly for the singular part. For the digital map φ+, the cocycle is φ+(σx) = β(φ+(x) − x_1). Hence x ∈ (φ+)^{-1}(D) implies σx ∈ (φ+)^{-1}(D) only if β(D − A) ⊆ D; this is neither stated nor proved, and is generally false (e.g., in Example 1, β·1 − 0 = β is not an atom). The same obstruction applies to the singular-support set S. Consequently the 0–1 law argument does not go through and purity is not established. The reader's separate concern about Theorem 5—that Lebesgue ψ need not imply absolute continuity of ν—does not land: any probability measure on R^r whose mod 1 projection is Lebesgue is absolutely continuous, by disintegrating into atomic fiber measures, each a countable sum of translated Lebesgue pieces. Thus the actual load-bearing gap is Lemma 1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies measures obtained by pushing forward the Parry measure on a primitive sofic shift K+ under the digital map x ↦ Σ_{k≥1} x_k β^{-k}, where β > 1 is a Pisot number and the digits belong to a finite integer alphabet. The author states four main results: a purity theorem for the pushed measures (Theorem 1), a finite/perfect dichotomy for the image set φ+(K+) (Theorems 2 and 3), a formula expressing lim_{k→∞} ν̂(zβ^k) as a Fourier coefficient of a two-sided torus measure ψ (Theorem 4), and an equivalence between absolute continuity of ν and the vanishing of those limits for all nonzero z ∈ Z[β] (Theorem 5). Several examples illustrate atomic, absolutely continuous, and singular cases, including classical Erdős measures and greedy β-expansions.","tokens_in":13024,"tokens_out":28205,"duration_ms":294817,"significance":"If the results were correct, the paper would give an attractive automata-theoretic and Fourier characterization of absolute continuity for a broad class of digit measures, extending Erdős's classical work and paralleling recent Rajchman-property results for self-similar measures. The matrix-product point of view and the use of Pisot conjugates to lift the problem to a torus are natural and potentially useful, and the statement of Theorem 5 is elegant. The examples are informative and include both known and new-looking cases. However, several proofs contain load-bearing errors, so the main theorems are not established as written.","major_comments":[{"comment":"The proof of Lemma 1 claims that the sets X^{-1}(D) and X^{-1}(S) are shift-invariant. This is false. Since X∘σ = X − X_1, or equivalently X∘σ = β(X − X_1) for the digital map, the membership of σx in X^{-1}(D) requires X(x) − X_1(x) ∈ D, which is an invariance property of the atom set that is neither stated nor proved. In the setting of Example 1, D = {0, ±1, ±1/β} and A = {−1, 0, 1}; with digit a = 0 and atom 1 ∈ D, the required condition would give β(1 − 0) = β ∉ D. Thus the ergodicity argument does not go through, and Theorem 1 is unsupported. A different proof of purity, or a corrected hypothesis that guarantees the needed invariance, is required.","section":"§5, Lemma 1"},{"comment":"Lemma 4 asserts that for an atom x of ν, the equality (φ+)^{−1}({x}) = (φ+)^{−1}(x−ε, x+ε) holds for small ε because the set of atoms is finite. Finiteness separates x from the other atoms, but not from the continuous part of the image: if x is not an isolated point of φ+(K+), every interval around x contains image points different from x. The asserted equality would in fact force x to be isolated in φ+(K+), which is precisely the dichotomy that Theorem 2 is trying to establish. Consequently the use of Lemma 4 in the proof of Theorem 2 to exclude atoms in the perfect case is invalid.","section":"§5, Lemma 4"},{"comment":"The proof of Lemma 3 claims that every element of E is an algebraic integer. This is false for β = 2, which the paper explicitly treats as a Pisot number. With digits {0, ±1}, the word (1, −1, −1) satisfies 1/2 − 1/4 − 1/4 = 0, and the suffix after the first digit has value −1/2, which is not an algebraic integer. The bounded-conjugate argument therefore does not apply as written. The lemma may still be true, but the finiteness proof needs a different mechanism, such as a finite carry set, before the statement can be accepted.","section":"§5, Lemma 3"},{"comment":"The proof of Theorem 5 asserts without argument that if ψ is Lebesgue measure on T^r, then ν = P_*∘~Φ_*(μ) is absolutely continuous on R. This implication is true, but it is not immediate and the manuscript should prove it. One natural proof is to disintegrate η = ~Φ_*(μ) with respect to Lebesgue measure on T^r along the fibers of the mod-1 map; since these fibers are copies of Z^r, a finite measure on R^r whose mod-1 projection is Lebesgue is a countable sum of absolutely continuous pieces. As written, a central step of the characterization is missing, although it is repairable.","section":"§6, Theorem 5"}],"minor_comments":[{"comment":"There are typographical errors, for example 'refe r' in the first paragraph of Section 1.","section":"Introduction"},{"comment":"In the proof of Lemma 2, the sentence 'from which shows that x ∈ Q(β)' is ungrammatical; moreover, the return time n depends on the chosen y, and the paper should state explicitly that the conjugate bound remains uniform because |β_q|^n ≤ 1 for |β_q| < 1.","section":"§5, Lemma 2"},{"comment":"Example 4 quotes a numerical value of lim_{n→∞} ν̂(β^n) without specifying the accuracy or the method of computation; a brief description of the numerical procedure and an error bound would be helpful.","section":"§7, Example 4"},{"comment":"Remark 4 states that B is ergodic with respect to ψ because ψ is the pushforward of an ergodic measure under Φ; this is correct, but the manuscript should also justify that Φ intertwines the shift with B, since Φ is not known to be injective.","section":"§6, Remark 4"},{"comment":"Reference [5] is cited as forthcoming; if it has appeared by the time of publication, the citation should be updated.","section":"References"}],"recommendation":"reject","confidential_remarks":"The problematic steps are mathematical rather than stylistic: Lemma 1 and Lemma 4 are false as stated, and they underpin the two main structural theorems. The proof of Lemma 3 also contains a false algebraic-integer claim, and Theorem 5 skips a non-obvious step, although that step is true and fixable. A substantially rewritten version with a new purity argument could be reconsidered, but in its present form the paper's central claims are not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Notable extension of Erdős measure theory to arbitrary primitive sofic shifts. The paper gives an automaton criterion for finiteness of the image, a purity claim, and a Fourier-limit criterion for absolute continuity. The examples (golden ratio expansions, redundant binary) are well chosen and genuinely informative.\n\nThe main soft spot is the proof of purity. Lemma 1 claims that for an ergodic shift and X = Σ x_n/β^n, the preimage of the atomic support D is shift-invariant. That is false. From X(σx) = β(X(x) - x_1), if X(x) ∈ D and x_1 = a, then X(σx) = β(y-a). For this to lie in D for all such x, you need β(D-A) ⊆ D, which is neither true in general nor proved. The same issue applies to the singular-support set. So the 0-1 argument collapses and Theorem 1 is not established as written.\n\nThe stress-test is right that the reader's separate objection to Theorem 5 is off the mark. If ρ is the linear map to the torus, it is invertible; ψ Lebesgue implies ρ_*μ' is a.c., hence μ' is a.c., hence its projection ν is a.c. The proof as written skips this, but it can be filled in. So the Rajchman-type criterion is on much firmer ground.\n\nEverything else is standard and careful: Perron-Frobenius, matrix product Fourier expressions, the finite-atomic case bounded by vertex count. No circularity, no fitted parameters. The paper is honest and readable.\n\nBottom line: the central purity theorem currently rests on an invalid lemma. That is a load-bearing gap. But it is not obviously unrecoverable; the specific cocycle relation may force purity via a different argument. I would send it to a knowledgeable referee, asking them to focus on Lemma 1 and whether the theorem can be rescued. If it cannot, the paper still contains useful tools and examples, but the main claim needs to be withdrawn or qualified.","headline":"The purity theorem's proof has a real gap in Lemma 1, but the paper extends Erdős measures to sofic shifts in a way that deserves a serious referee.","tokens_in":13625,"tokens_out":15380,"would_cite":false,"duration_ms":155944,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A63","68Q45","11K55","37A45"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that digit-expansion measures from sofic shifts are pure, and that absolute continuity is equivalent to vanishing of all Fourier limits at Pisot powers.","keywords":["sofic shifts","Pisot numbers","Erdős measures","Parry measure","absolute continuity","Fourier transform","digital expansions","Jessen-Wintner theorem"],"falsifier":"Construct a primitive finite automaton and a Pisot $\\beta$ for which all limits $\\lim_{k\\to\\infty}\\hat\\nu(z\\beta^k)$ with $z\\in\\mathbb{Z}[\\beta]\\setminus\\{0\\}$ vanish but $\\nu$ is singular (e.g. singular continuous); such an example would refute Theorem 5. Concretely, one could search among automata of the same small size as the paper's examples, computing the matrix product numerically, and then test absolute continuity by estimating the density of $\\nu$ from long digit strings.","tokens_in":12527,"feed_emoji":"🔢","tokens_out":8257,"duration_ms":81108,"temperature":0.7,"pith_summary":"This paper studies probability measures obtained by picking digit strings from a finite automaton, weighting them by the maximum-entropy (Parry) measure, and evaluating the infinite series $\\sum x_k\\beta^{-k}$ for a Pisot number $\\beta>1$. Such measures generalize the classical Erdős measures that arise from Bernoulli convolutions, and they appear in numeration systems, tilings, and spectral theory. The paper establishes two structural results: every such measure is pure (atomic, singular continuous, or absolutely continuous), and absolute continuity is equivalent to the vanishing of the Fourier transform along the scaled sequence $z\\beta^k$ for all nonzero $z\\in\\mathbb{Z}[\\beta]$. This gives a finite, automaton-checkable criterion for deciding whether a digit system produces a smooth measure, and it recovers known cases such as greedy golden-ratio expansions being Lebesgue and redundant base-2 expansions being singular.","feed_headline":"Fourier test decides when digit measures are smooth","feed_subtitle":"For Pisot-base digit systems, one Fourier-limit condition separates continuous from singular measures.","key_machinery":"The load-bearing device is the two-sided digital map $\\Phi:K\\to\\mathbb{T}^r$, sending a bi-infinite digit sequence to the vector of fractional parts $(\\sum_{k=-\\infty}^{\\infty} x_k\\beta^{-k+m})_{m=0}^{r-1}$; the Pisot property makes the series converge and makes this map conjugate the shift to a hyperbolic toral endomorphism. Its push-forward $\\psi$ has Fourier coefficient at $(m_0,\\dots,m_{r-1})$ equal to $\\lim_{k\\to\\infty}\\hat\\nu(z\\beta^k)$ for $z=m_0+\\cdots+m_{r-1}\\beta^{r-1}$ (Theorem 4). The Fourier transform of $\\nu$ is also written as an infinite product of weighted transition matrices, $\\hat\\nu(t)=v_L^T\\prod_{n=1}^\\infty W(\\beta^{-n}t)\\,v_R$ with $W(t)=\\lambda^{-1}\\sum_a e(-at)M_a$, which is what makes the limits computable in examples.","core_discovery":"The paper's central claim is that for every Pisot number $\\beta$ and every sofic shift with primitive language, the measure $\\nu$ obtained by pushing the Parry measure through the digital map $(x_k)\\mapsto\\sum_{k=1}^\\infty x_k\\beta^{-k}$ is absolutely continuous with respect to Lebesgue measure if and only if $\\lim_{k\\to\\infty}\\hat\\nu(z\\beta^k)=0$ for every nonzero $z\\in\\mathbb{Z}[\\beta]$ (Theorem 5). The 'if' direction is proved by packaging all these limits as the Fourier coefficients of a measure $\\psi$ on the $r$-dimensional torus obtained from the two-sided shift: $\\psi$ is Lebesgue exactly when all coefficients vanish, and the paper asserts that Lebesgue $\\psi$ forces $\\nu$ to be absolutely continuous. The 'only if' direction is the standard fact that a non-vanishing limit obstructs absolute continuity. The same framework shows the measure is pure (Theorem 1) and that it is atomic exactly when every cycle in the automaton has the same digit-series value (Theorems 2 and 3).","pith_inferences":["A practical test emerges that the author did not spell out: enumerate a finite set of $z\\in\\mathbb{Z}[\\beta]$ and approximate $\\lim_k\\hat\\nu(z\\beta^k)$ by truncating the matrix product; any nonzero limit certifies singularity, and vanishing on the finite set is evidence for absolute continuity.","The unproven implication in Theorem 5's proof—Lebesgue $\\psi$ forces absolutely continuous $\\nu$—is the natural place to look for a hidden counterexample; testing it requires a sofic shift whose torus measure is uniform while its one-dimensional projection is singular.","The same matrix-product formalism should extend to non-Parry weightings (e.g. arbitrary Bernoulli digit probabilities), where purity may fail but the Fourier-limit criterion for absolute continuity may still hold in modified form.","Combining purity with Hausdorff dimension computations could yield that dimension 1 plus vanishing limits implies absolute continuity, a sharper statement than either property alone."],"forward_implications":["For any sofic digit system with Pisot base, the question 'is the measure smooth?' reduces to checking whether certain Fourier limits vanish; this is in principle decidable by approximating matrix products.","The purity theorem rules out mixtures: a measure of this type can never be partly absolutely continuous and partly singular.","The atomic case has a graph-theoretic characterization: the measure is a finite sum of point masses exactly when every cycle in the automaton satisfies equation (17) with the same value.","The class covered includes measures from optimal base-2 expansions used in fast scalar multiplication on elliptic curves and spectral measures of substitution systems, so the criterion applies across those settings.","If the main theorem is right, then for this class the Rajchman property—Fourier transform tending to 0 at infinity—actually implies absolute continuity, matching a recently proved phenomenon for self-similar measures."],"supporting_citations":[{"why":"Erdős's first Bernoulli convolution paper; it supplies the motivating measures and the method of proving singularity from non-decay of Fourier transforms.","marker":"[12]"},{"why":"Erdős's extension to all irrational Pisot numbers; the template for the Pisot-power Fourier argument.","marker":"[13]"},{"why":"Parry's β-expansions paper; establishes that Pisot β give sofic β-shifts, the setting for the digital maps.","marker":"[26]"},{"why":"Defines the Parry (measure of maximal entropy) on shifts of finite type; used as the canonical measure on K+.","marker":"[27]"},{"why":"Jessen–Wintner theorem; the source of the purity result that Lemma 1 adapts to ergodic shift-invariant measures.","marker":"[23]"},{"why":"Sidorov–Vershik study of the Erdős measure via a two-sided map to a torus; the basis for Theorem 4's interpretation of Fourier limits.","marker":"[38]"},{"why":"Recent result on self-similar measures showing the Rajchman property is equivalent to absolute continuity; the analogue the paper points to for Theorem 5.","marker":"[5]"},{"why":"Supplies Example 5, the optimal base-2 representation measure from cryptography, which the paper's results classify as singular.","marker":"[17]"}],"fun_headline_variants":["Fourier limit marks smooth digit measures","Pisot digit measures: one test tells smooth from singular","When do digit measures become Lebesgue? A Fourier answer","Purity via Fourier: digit measures split cleanly","Smooth or singular? Fourier decides for Pisot digit systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's key unproven premise is that if the auxiliary measure on the high-dimensional torus is Lebesgue measure, then the projected one-dimensional measure $\\nu$ must be absolutely continuous; singular measures on the line can project to Lebesgue on the circle, so this implication needs the special structure of these digital measures and is not established in the paper.","fun_headline_variants_meta":{"raw":{"variants":["Fourier limit marks smooth digit measures","Pisot digit measures: one test tells smooth from singular","When do digit measures become Lebesgue? A Fourier answer","Purity via Fourier: digit measures split cleanly","Smooth or singular? Fourier decides for Pisot digit systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000393,"raw_usage":{"total_tokens":2002,"prompt_tokens":821,"completion_tokens":1181,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":437,"completion_tokens_details":{"reasoning_tokens":1102}},"tokens_in":437,"tokens_out":1181,"duration_ms":8913,"temperature":1.0,"reasoning_tokens":1102,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:26:21.307609+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a primitive finite automaton and a Pisot $\\beta$ for which all limits $\\lim_{k\\to\\infty}\\hat\\nu(z\\beta^k)$ with $z\\in\\mathbb{Z}[\\beta]\\setminus\\{0\\}$ vanish but $\\nu$ is singular (e.g. singular continuous); such an example would refute Theorem 5. Concretely, one could search among automata of the same small size as the paper's examples, computing the matrix product numerically, and then test absolute continuity by estimating the density of $\\nu$ from long digit strings.","supporting_citations":[{"cited_title":"Erd˝ os,On a family of symmetric Bernoulli convolutions , Amer","cited_arxiv_id":null,"evidence_quote":"Erdős's first Bernoulli convolution paper; it supplies the motivating measures and the method of proving singularity from non-decay of Fourier transforms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Erdős's extension to all irrational Pisot numbers; the template for the Pisot-power Fourier argument."},{"cited_title":"Parry, On the β -expansions of real numbers , Acta Math","cited_arxiv_id":null,"evidence_quote":"Parry's β-expansions paper; establishes that Pisot β give sofic β-shifts, the setting for the digital maps."},{"cited_title":"Parry, Intrinsic Markov chains , Trans","cited_arxiv_id":null,"evidence_quote":"Defines the Parry (measure of maximal entropy) on shifts of finite type; used as the canonical measure on K+."},{"cited_title":"Jessen and A","cited_arxiv_id":null,"evidence_quote":"Jessen–Wintner theorem; the source of the purity result that Lemma 1 adapts to ergodic shift-invariant measures."},{"cited_title":"Sidorov and A","cited_arxiv_id":null,"evidence_quote":"Sidorov–Vershik study of the Erdős measure via a two-sided map to a torus; the basis for Theorem 4's interpretation of Fourier limits."},{"cited_title":"Self-similar measures and the Rajchman property","cited_arxiv_id":"1910.03463","evidence_quote":"Recent result on self-similar measures showing the Rajchman property is equivalent to absolute continuity; the analogue the paper points to for Theorem 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Example 5, the optimal base-2 representation measure from cryptography, which the paper's results classify as singular."}],"review_version":1}