{"id":"f8767920-745a-4833-b2b8-37bf89a3d2b2","arxiv_id":"2510.08051","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For invariant measures on the Hilbert cube shift, mean Rényi information dimension equals information dimension rate, and several new rate-distortion entropies are shown to coincide with Kolmogorov–Sinai entropy.","lead":"This paper proves that for shift-invariant (possibly non-ergodic) measures on the Hilbert cube, the mean Rényi information dimension equals the information dimension rate, answering an open question from Gutman and Śpiewak. It also introduces four rate-distortion entropies and shows they coincide with Kolmogorov–Sinai entropy under broad assumptions, and extends the double variational principle for mean dimension to ergodic measures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main gap: Step 1 of Theorem 1.2 extends [W21, Prop. 4.2] to all invariant measures on the strength of a footnote; without that extension the non-ergodic rate-distortion entropy equality is unsupported.","rationale":"I read the paper in good faith. Theorem 1.1, which answers Gutman–Śpiewak's Problem 2, appears substantiated: the proof combines Geiger–Koch's identification of information dimension rate with L2 rate-distortion dimension, the two-sided inequality of [GS20], and Lemma 3.1; I did not find a flaw in the constant chasing or the partition-entropy identities. The load-bearing weakness is exactly the one the reader identified: the extension of [W21, Proposition 4.2] from ergodic to invariant measures is asserted in a footnote with no adaptation. This extension is needed in Section 3.2, Step 1, to prove that the Pfister–Sullivan ε-entropy, for arbitrary invariant μ, is bounded above by the L1 rate-distortion entropy; without it, Theorem 1.2(2) for non-ergodic measures is not proved. I do not think this is a manufactured objection: the manuscript itself flags the missing proof. I also considered whether Theorem 1.3 contains an internal inconsistency; I found the ergodic supremum step plausible, with the main external input being [YCZ25, Lemma 2.3(3)], which is published and therefore a legitimate black box, though its use should be checked. Because the reader's conditional verdict already reflects the same central concern, my stress-test does not change the verdict; if the author supplies the missing invariant-measure proof or a reference, the paper should be accepted.","tokens_in":19263,"tokens_out":25637,"duration_ms":187292,"concrete_test":"Re-derive [W21, Proposition 4.2] for invariant measures with ergodic decomposition μ = ∫ m dτ(m). Check whether the inequality can be proved by applying the ergodic version to each component and integrating, i.e., whether inf_F limsup (1/n) log s(X_{n,F}, d_n, ε) is bounded by L/(L−1) ∫ R_{m,L1}(1/(6L+2)ε) dτ(m), and whether this integral is dominated by R_{μ,L1}(·). If this chain fails for a two-component mixture, for example on a full shift with μ = (1/2)(Bernoulli(1/3) + Bernoulli(2/3)), then the asserted extension is false and Theorem 1.2(2) needs a separate argument. If the chain holds, the gap is fillable and Theorem 1.2(2) stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step is Section 3.2, Step 1, inequality (3.7). For a general invariant measure μ, the author invokes [W21, Proposition 4.2] to conclude that lim_{ε→0} PS_μ(T,ε) ≤ h_{μ,L1}(T). Footnote 4 concedes that the proposition is stated for ergodic measures and asserts, without proof, that 'the proof applies to invariant measures.' If this extension is false, the chain lim PS_μ ≤ h_{μ,L1} ≤ h_μ is broken; Lemma 3.4(2) gives h_μ = lim PS_μ only under the g-almost product property, so equality in Theorem 1.2(2) would fail for non-ergodic μ. The issue is not cosmetic: the original proposition's proof plausibly uses ergodicity (e.g., constancy of local entropy/Shannon–McMillan–Breiman), and the author's Step 1 applies the extension globally, even outside the g-almost-product class. Theorem 1.1 is unaffected: its proof is self-contained modulo Geiger–Koch and [GS20, Prop. C-B.1]. The reader's secondary concern about Lemma 2.3(3) is less serious, since [YCZ25] is a published external result rather than an unproved internal extension.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper revisits the connection between mean dimension and rate-distortion theory. Theorem 1.1 proves that for every invariant measure on the Hilbert cube shift ([0,1]^Z, σ, d_Z), the lower/upper mean Rényi information dimension equals the lower/upper information dimension rate, answering a question of Gutman and Śpiewak. Theorem 1.2 introduces L_p, L∞, Bowen, and r rate-distortion entropies and proves their coincidence with Kolmogorov–Sinai entropy, for all ergodic measures and, under the g-almost product property, for all invariant measures. Theorem 1.3 establishes, for systems with the marker property and finite mean dimension, a double variational principle in which the supremum is taken over ergodic measures, for a family E of measure-theoretic ε-entropies and for L_p rate-distortion functions. The final section discusses applications to lower Brin–Katok entropy and to exchanging limsup and supremum.","tokens_in":19610,"tokens_out":14679,"duration_ms":117460,"significance":"If correct, the paper resolves a known open problem and gives a systematic comparison of rate-distortion entropies with classical entropy, together with an ergodic version of the double variational principle. The proof of Theorem 1.1 is mostly self-contained modulo standard external results and is clearly presented, as is the combinatorial comparison between Bowen and mean metrics in Theorem 1.2. However, the non-ergodic part of Theorem 1.2(2) relies on an unproved extension of a result of T. Wang, and Theorem 1.3 depends heavily on Lemma 2.3(3) imported from the author's previous work, so the incremental contribution is partly based on black-boxed ingredients.","major_comments":[{"comment":"The proof of Theorem 1.2(2) for non-ergodic measures hinges on applying [W21, Proposition 4.2] to every invariant measure. Footnote 4 states that the proposition is proved for ergodic measures but 'the proof applies to invariant measures,' without giving the adaptation. This is load-bearing: inequality (3.7) is the only lower bound relating Pfister–Sullivan ε-entropy to the L_p rate-distortion entropy, and Lemma 3.4(2) supplies the matching upper bound only under the g-almost product property. If the asserted extension fails, the equalities in Theorem 1.2(2) are unsupported for non-ergodic μ. Please provide a full proof of the extension or explicitly restrict the statement to ergodic measures.","section":"§3.2, Step 1 (just after Eq. (3.8))"},{"comment":"Lemma 2.3(3), quoted from the author's own [YCZ25], is used as a black box at decisive points: Lemma 3.7(1), Theorem 1.3 Step 1, and Theorem 1.3 Step 2. It contains exactly the candidate-independence and variational-principle statements that the paper claims to revisit. Because of this dependence, the manuscript should either reproduce a proof, give precise theorem numbers and hypotheses from [YCZ25], or state clearly that Theorem 1.3 is a corollary of [YCZ25, Theorems 1.1–1.3] plus the ergodic-restriction argument. As written, the novelty boundary between this paper and [YCZ25] is difficult to assess.","section":"§2.3, Lemma 2.3(3)"},{"comment":"The proof of Step 1 is written for h_μ(T,ε) ∈ E, but the theorem claims the inequality for h_μ(T,ε) ∈ E ∪ {R_{μ,L_p}}. The inclusion of R_{μ,L_p} is not justified in Step 1; it appears only in Step 2 via a brief '≤' remark. The missing argument is the elementary inequality R_{μ,L_1}(ε) ≤ R_{μ,L_p}(ε) for p ≥ 1, which would give sup_E rdim_{L1}(d) ≤ sup_E rdim_{Lp}(d). Please add this explicitly or adjust the theorem statement accordingly.","section":"§3.3, Step 1 of Theorem 1.3"}],"minor_comments":[{"comment":"Typo: 'four types pf rate-distortion entropies' should be 'four types of rate-distortion entropies'.","section":"§1, Introduction"},{"comment":"The same symbol d_n is used for the Bowen metric and the mean metric. Using \\bar d_n consistently for the mean metric would remove ambiguity in the proof of (3.8).","section":"§3.2"},{"comment":"The displayed equality lim_{r→0} h_{μ,r}(T) = sup_{r>0} sup_{ε>0} R_{μ,r}(ε) = lim_{ε→0} lim_{r→0} R_{μ,r}(ε) involves an interchange of limits that should be justified by the monotonicity of R_{μ,r}(ε) in ε and r, since it is not immediate.","section":"§3.2, Step 2"},{"comment":"The reference to [YCZ25] should name the specific theorems (e.g., Theorems 1.1–1.3) in the bibliography entry, since Lemma 2.3(3) packages several substantial statements whose exact hypotheses matter for the present paper.","section":"§2.3, Lemma 2.3(3)"},{"comment":"Typo: 'there exists exists ε_0' should be 'there exists ε_0'.","section":"§3.4, Theorem 3.9"}],"recommendation":"major_revision","confidential_remarks":"The main concern for the editor is that Theorem 1.2(2), a central advertised result, depends on an unproved extension of [W21, Proposition 4.2] to all invariant measures. This is fixable either by supplying the proof or by restricting the scope, but it needs to be addressed before the paper can be accepted. The heavy reliance on the author's own [YCZ25] is also worth monitoring: while citing published work is legitimate, the paper should make the import explicit so that the incremental contribution is transparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline first: the paper answers a genuinely open question from Gutman–Śpiewak, and for that result alone it should be read carefully. Theorem 1.1 extends the equality between mean Rényi information dimension and information dimension rate from ergodic to all invariant measures on the Hilbert cube shift. The proof is clean and self-contained modulo Geiger–Koch and the GS20 comparison inequality, and Lemma 3.1 is the right new tool. That is a real result, not a repackaging.\n\nThe paper also introduces four rate-distortion entropies and shows they coincide with Kolmogorov–Sinai entropy under the g-almost product property, and it widens the double variational principle to ergodic measures for a large candidate family. That is useful synthesis. The candidate set E is broad, and the Fatou argument in Lemma 3.7 is a neat way to reduce the supremum to ergodic measures.\n\nNow the soft spot, and the reader's report is right about it. In Step 1 of Theorem 1.2 the proof invokes [W21, Proposition 4.2] for all invariant measures. That proposition is stated for ergodic measures. Footnote 4 says 'the proof applies to invariant measures' but no adaptation is shown. Ergodicity can matter in the local-entropy estimates, so the assertion is not obviously harmless. The chain in (3.7) feeding Lemma 3.4(2) rests on it. If the extension is false, the non-ergodic equality in Theorem 1.2(2) is unsupported. I could not verify the extension from the paper, and this is the main thing a referee must pin down. The theorem should be treated as conditional rather than fully proven. Theorem 1.1 does not depend on this shaky step.\n\nThe secondary concern—Lemma 2.3(3) quoted from the author's own [YCZ25]—is weaker. It is a published external result, the citation is explicit, and self-citation is not itself a flaw. Theorem 1.3 does depend on it, but that is a normal dependence on prior work, not an internal gap.\n\nWho should read this: anyone working on mean dimension, rate-distortion theory, or measure-theoretic ε-entropies. The main theorem is solid; the rest is valuable but needs a supporting proof. Yes, send it to peer review. Ask for the invariant-measure version of [W21, Prop. 4.2] to be written out, or restrict Theorem 1.2(2) to the ergodic case if the extension fails. If that piece holds, the paper is a solid contribution.","headline":"A real answer to an open question, wrapped in a broader unification that has one load-bearing gap: the non-ergodic extension of Wang's proposition is asserted, not proved.","tokens_in":20107,"tokens_out":3015,"would_cite":true,"duration_ms":26328,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A05","37A35","37B40","94A34"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for every invariant measure on the Hilbert cube shift, the lower and upper mean Rényi information dimensions coincide with the corresponding information dimension rates, answering an open question from earlier work, an","keywords":["mean dimension","rate-distortion functions","mean Rényi information dimension","information dimension rate","Kolmogorov–Sinai entropy","variational principle","ergodic measures","marker property"],"falsifier":"Compute the L∞ rate-distortion entropy lim_{ε→0} R_{μ,L∞}(ε) for a non-ergodic measure, say μ = (1/2)δ_{0^∞} + (1/2)L^Z on [0,1]^Z, and compare it with h_μ(σ). If they differ, Theorem 1.2(2) fails for non-ergodic measures. For Theorem 1.1, compute MRID for the same μ via optimal ε-partitions and compare with d(μ); a mismatch would disprove the claimed equality.","tokens_in":19120,"feed_emoji":"📏","tokens_out":5237,"duration_ms":43823,"temperature":0.7,"pith_summary":"This paper's central claim is that two seemingly different ways of measuring the 'information dimension' of an invariant measure on the Hilbert cube shift—one based on optimally fine partitions, the other on coordinate quantizers—always agree, even when the measure is not ergodic. This settles an open question that had previously been resolved only for ergodic measures. The paper also introduces four types of rate-distortion entropies and proves that, under reasonably general shadowing conditions, each converges to the classical Kolmogorov–Sinai entropy of the measure. Finally, for systems with the marker property and finite mean dimension, it shows that the double variational principle for mean dimension holds with the supremum taken over ergodic measures, and that this extends to a whole family of measure-theoretic ε-entropies. The upshot is a unified picture in which rate-distortion, partition entropy, and information dimension give compatible answers about the complexity of infinite-entropy systems.","feed_headline":"Mean Rényi dimension equals information rate for all measures","feed_subtitle":"Two independent definitions of dynamical dimension agree without ergodicity, closing a question from earlier work.","key_machinery":"The load-bearing object is the mean Rényi information dimension, MRID(X,T,d,μ)=liminf_{ε→0} (inf_{diam(α)≤ε} h_μ(T,α))/log(1/ε), compared with the information dimension rate defined through coordinate quantizer partitions α_m. The proof of Theorem 1.1 bridges the two through L^2 rate-distortion functions: known results identify the information dimension rate with the L^2 rate-distortion dimension, and a new inequality (Lemma 3.1) shows rdim_{L^p} ≤ MRID, while a covering argument yields the reverse bound. For Theorem 1.2, an empirical-measure ε-entropy serves as the bridge: the proof establishes an inequality chain from that entropy to the rate-distortion entropies and then to Kolmogorov–Sin","core_discovery":"The main result is Theorem 1.1: for every σ-invariant Borel probability measure μ on [0,1]^Z with the metric d_Z(x,y)=Σ_{n∈Z}|x_n−y_n|/2^{|n|}, the lower and upper mean Rényi information dimensions MRID and MRID equal the lower and upper information dimension rates d(μ) and d̄(μ). This was known for ergodic measures; the paper removes the ergodicity assumption. Theorem 1.2 introduces L^p, L^∞, Bowen, and r rate-distortion entropies and shows that for ergodic measures all equal h_μ(T), while under the g-almost product property this equality holds for every invariant measure. Theorem 1.3 states that if a system has the marker property and finite mean dimension, the double variational principle","pith_inferences":["The equality in Theorem 1.1 likely extends to shifts over arbitrary compact metric alphabets, since the proof only uses rate-distortion comparisons and the shift structure; this could be checked directly for product measures on X^Z.","The same proof technique may yield a shorter route to the known ergodic case and clarify when the lower and upper versions coincide or differ.","If the cited inequality for non-ergodic measures used in Theorem 1.2(2) is not reproved in that generality, the non-ergodic part of that theorem rests on an unverified extension; a counterexample would leave only the ergodic statement intact.","The marker-property result suggests that the maximality of ergodic measures for metric mean dimension is a general phenomenon under finite mean dimension, rather than an exceptional one."],"forward_implications":["For every invariant measure on the Hilbert cube shift, partition-based and quantizer-based notions of information dimension agree, so non-ergodic measures no longer require special treatment.","The four new rate-distortion entropies give information-theoretic characterizations of Kolmogorov–Sinai entropy, meaning rate-distortion theory and classical entropy coincide in these settings.","Under the marker property with finite mean dimension, the supremum in the double variational principle can always be restricted to ergodic measures, sharpening the previously known variational formulation.","The equality extends to a wide menu of measure-theoretic ε-entropies, indicating that the choice of which ε-entropy one uses does not change the metric mean dimension for these systems.","Systems with vanishing Kolmogorov–Sinai entropy, such as translations of compact groups, automatically have zero rate-distortion entropies of all four new types."],"fun_headline_variants":["Mean Rényi dimension equals rate without ergodicity","Dimension equality answers Gutman-Śpiewak question","Four rate-distortion entropies match Kolmogorov-Sinai","Marker systems shrink variational principle to ergodic measures"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of part (2) of Theorem 1.2 assumes, without adaptation, that an inequality previously proved for ergodic measures also holds for all invariant measures; if that extension is false, the claim that the new rate-distortion entropies equal Kolmogorov–Sinai entropy for non-ergodic measures loses its support.","fun_headline_variants_meta":{"raw":{"variants":["Mean Rényi dimension equals rate without ergodicity","Dimension equality answers Gutman-Śpiewak question","Four rate-distortion entropies match Kolmogorov-Sinai","Marker systems shrink variational principle to ergodic measures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000402,"raw_usage":{"total_tokens":1927,"prompt_tokens":733,"completion_tokens":1194,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":1126}},"tokens_in":477,"tokens_out":1194,"duration_ms":10570,"temperature":1.0,"reasoning_tokens":1126,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T10:49:57.338862+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the L∞ rate-distortion entropy lim_{ε→0} R_{μ,L∞}(ε) for a non-ergodic measure, say μ = (1/2)δ_{0^∞} + (1/2)L^Z on [0,1]^Z, and compare it with h_μ(σ). If they differ, Theorem 1.2(2) fails for non-ergodic measures. For Theorem 1.1, compute MRID for the same μ via optimal ε-partitions and compare with d(μ); a mismatch would disprove the claimed equality.","supporting_citations":[],"review_version":1}