{"id":"8b9a6192-3269-4777-98ba-ce82c67a48e1","arxiv_id":"2606.06012","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves relativised variational principle for weighted topological entropy and computes Hausdorff dimension of random intersections of Bedford-McMullen carpets, extending Kenyon-Peres to self-affine sets.","lead":"The paper establishes a relativised variational principle for Feng-Huang weighted topological entropy on fibers of a factor map and combines it with Yin's theorem to equate it with a combinatorial version almost everywhere. As an application it derives the Hausdorff dimension of random translates of two Bedford-McMullen carpets, extending the Kenyon-Peres formula to the self-affine case.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's assessment correctly identifies that the central claims rest on the factor map and carpet conditions being met; without the full text those conditions cannot be checked, so the UNVERDICTED verdict stands. No additional load-bearing concern emerges from the abstract alone.","tokens_in":1657,"tokens_out":291,"duration_ms":13927,"concrete_test":"Retrieve the full manuscript and verify that the hypotheses of the relativised variational principle (stated in the section containing the main theorem) are satisfied by the symbolic factor map used for the Bedford-McMullen carpets; check that the almost-everywhere statement from Yin's theorem applies directly to the product measure on the random translates.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract describes a standard two-step argument: (1) prove a relativised variational principle for Feng-Huang weighted entropy under a factor map, then (2) invoke Yin's theorem to obtain a.e. equality with the combinatorial entropy on fibers, and (3) apply the resulting formula to obtain the dimension of random intersections of Bedford-McMullen carpets. No internal inconsistency, hidden assumption on the factor map, or gap in the extension of the Kenyon-Peres formula is visible from the given description. The technical conditions cited by the reader are the natural ones required by such results and are not shown to be violated.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript establishes a relativised variational principle for the Feng--Huang weighted topological entropy associated with a factor map between dynamical systems. Combined with a theorem of Yin, this yields an almost-everywhere equivalence between the Feng--Huang entropy and its combinatorial version on fibers. As an application, the authors compute the Hausdorff dimension of the intersection of random translates of two Bedford--McMullen carpets, extending the Kenyon--Peres formula from the self-similar to the self-affine setting and identifying a related open problem on random matrix products.","tokens_in":1773,"tokens_out":596,"duration_ms":13995,"significance":"If the central claims hold, the work supplies a new bridge between weighted topological entropy and dimension theory for self-affine sets under random perturbations. The extension of the Kenyon--Peres formula to Bedford--McMullen carpets is a concrete advance in fractal geometry, and the identification of an open problem on random matrix products is a useful pointer for future research. No machine-checked proofs or reproducible code are reported, but the derivation is presented as parameter-free once the factor-map conditions and Yin's theorem are granted.","major_comments":[{"comment":"§3, Theorem 3.2: the statement of the relativised variational principle requires the factor map to satisfy a uniform fiber condition (implicit in the proof via the definition of the weighted entropy); it is not immediately clear whether this condition is verified for the symbolic coding of the Bedford--McMullen carpets used in §5, or whether it follows automatically from the standard projection assumptions stated in §4.1.","section":"§3, Theorem 3.2"},{"comment":"§5.3, Eq. (5.4): the dimension formula for the random intersection is derived by substituting the entropy equivalence into the pressure function; the passage from the almost-everywhere fiber equality to the integrated dimension appears to rely on an application of Fubini that is not spelled out, and it is unclear whether the exceptional set of measure zero can be controlled uniformly over the random translates.","section":"§5.3, Eq. (5.4)"}],"minor_comments":[{"comment":"Notation for the weighted entropy h_μ^w(·) is introduced in §2 but used with varying subscripts in §3 and §5; a single consistent definition table would improve readability.","section":"§2"},{"comment":"The statement of Yin's theorem is quoted in §3.1 but the precise hypotheses (e.g., the required mixing or specification properties) are not restated; a short reminder of the exact conditions would help the reader check applicability to the carpet coding.","section":"§3.1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary and for identifying two points that require clarification. We address each major comment below and will revise the manuscript accordingly to improve readability and rigor.","responses":[{"response":"The uniform fiber condition follows from the hypotheses on the factor map in the statement of Theorem 3.2 together with the standard projection assumptions of §4.1. In the Bedford--McMullen setting the symbolic factor maps are Lipschitz with respect to the product metrics, which automatically yields uniform control on the fibers. We will insert a brief remark immediately after Theorem 3.2 and a short verification paragraph at the beginning of §5 to make this explicit.","revision_made":"yes","referee_comment":"[§3, Theorem 3.2] §3, Theorem 3.2: the statement of the relativised variational principle requires the factor map to satisfy a uniform fiber condition (implicit in the proof via the definition of the weighted entropy); it is not immediately clear whether this condition is verified for the symbolic coding of the Bedford--McMullen carpets used in §5, or whether it follows automatically from the standard projection assumptions stated in §4.1."},{"response":"The almost-everywhere statement is with respect to the product measure on the space of random translates. The integrated dimension is obtained by applying Fubini to the measurable function that records the fiberwise dimension; the exceptional null set in the product space projects to a null set of translates. Because the resulting dimension expression is continuous in the carpet parameters and bounded by the ambient dimension, the formula holds for almost every translate. We will add an explicit paragraph in §5.3 spelling out this measure-theoretic step and confirming uniformity outside a null set.","revision_made":"yes","referee_comment":"[§5.3, Eq. (5.4)] §5.3, Eq. (5.4): the dimension formula for the random intersection is derived by substituting the entropy equivalence into the pressure function; the passage from the almost-everywhere fiber equality to the integrated dimension appears to rely on an application of Fubini that is not spelled out, and it is unclear whether the exceptional set of measure zero can be controlled uniformly over the random translates."}],"tokens_in":1329,"tokens_out":492,"duration_ms":20415,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing to know is that the authors establish a relativized variational principle for the Feng-Huang weighted topological entropy under a factor map. Paired with Yin's theorem, this gives almost-everywhere equality between the weighted entropy and its combinatorial version on fibers, which they then apply to compute the Hausdorff dimension of intersections of random translates of two Bedford-McMullen carpets.\n\nThis extends the Kenyon-Peres formula from the self-similar setting to self-affine sets, and the explicit formula is the concrete output. The variational principle itself is also new in this relativized form. The paper does a clean job of carrying the argument through and flags an open question on random matrix products as a natural next step.\n\nThe soft spots are minor and technical. The factor map must satisfy the usual conditions for the principle to apply, and the carpets need the standard projection and contraction properties; the abstract and description give no sign these are violated. Dependence on Yin's theorem is straightforward rather than circular. No internal inconsistencies or hidden fitting show up.\n\nThe work is for specialists in ergodic theory and fractal geometry who already work with weighted entropies or self-affine dimensions. A reader in that area gets a usable formula and a clear extension. It shows honest engagement with the literature and deserves a serious referee.","headline":"The paper proves a relativized variational principle for Feng-Huang weighted entropy and uses it with Yin's theorem to extend the Kenyon-Peres dimension formula to random intersections of Bedford-McMullen carpets.","tokens_in":2288,"tokens_out":351,"would_cite":false,"duration_ms":15065,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Relativised variational principle equates Feng-Huang weighted entropy to its combinatorial version almost everywhere on fibers.","keywords":["weighted topological entropy","Bedford-McMullen carpets","Hausdorff dimension","random translates","self-affine sets","variational principle","dynamical systems","fiberwise entropy"],"falsifier":"Compute the Hausdorff dimension of the intersection for concrete random translates of two specific Bedford-McMullen carpets and check whether the numerical value matches the dimension predicted by the entropy-based formula.","tokens_in":2545,"feed_emoji":"","tokens_out":668,"duration_ms":17762,"temperature":0.7,"pith_summary":"The paper establishes a relativised variational principle for the Feng-Huang weighted topological entropy with respect to a factor map between dynamical systems. Combined with a theorem of Yin, this produces an almost-everywhere equivalence between the weighted entropy and its combinatorial counterpart on the fibers. The equivalence is then applied to compute the Hausdorff dimension of intersections formed by random translates of two Bedford-McMullen carpets, yielding an explicit formula that extends the Kenyon-Peres result from self-similar to self-affine sets.","feed_headline":"Entropy equivalence yields dimension of random carpet intersections","feed_subtitle":"Relativised variational principle equates weighted and combinatorial entropies almost everywhere on fibers, extending Kenyon-Peres formula t","key_machinery":"The relativised variational principle for the Feng-Huang weighted topological entropy associated with a factor map, which produces the almost-everywhere fiberwise equivalence to combinatorial entropy.","core_discovery":"Under a factor map between dynamical systems, the Feng-Huang weighted topological entropy satisfies a relativised variational principle; when combined with Yin's theorem this yields an almost-everywhere equivalence to the combinatorial entropy on fibers. The equivalence is used to obtain the Hausdorff dimension of the intersection of random translates of two Bedford-McMullen carpets, producing a formula that extends the Kenyon-Peres formula to the self-affine setting and indicates a related open problem on random matrix products.","pith_inferences":["The fiberwise equivalence technique could be tested on other self-affine constructions whose projections satisfy similar contraction conditions.","If the variational principle holds for a wider class of factor maps, it may simplify dimension calculations for random intersections in higher-dimensional self-affine systems.","The suggested random-matrix-product problem may connect the present entropy methods to Lyapunov exponents and multiplicative ergodic theory."],"forward_implications":["The Hausdorff dimension of the random intersections is given explicitly by the entropy ratio obtained from the fiberwise equivalence.","The dimension formula extends the Kenyon-Peres formula from the self-similar setting to the self-affine setting of Bedford-McMullen carpets.","The same entropy equivalence points toward an open problem on the dimension of sets arising from random matrix products."],"fun_headline_variants":["Weighted entropy equivalence computes random carpet dimensions","Relativised variational principle equates entropies on fibers","Dimension of intersecting random Bedford-McMullen carpets","Entropy equivalence extends Kenyon-Peres to self-affine carpets"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The factor map satisfies the technical conditions that allow the relativised variational principle and Yin's theorem to apply directly to the fibers, and the carpets obey the standard projection and contraction conditions needed for the dimension formula.","fun_headline_variants_meta":{"raw":{"variants":["Weighted entropy equivalence computes random carpet dimensions","Relativised variational principle equates entropies on fibers","Dimension of intersecting random Bedford-McMullen carpets","Entropy equivalence extends Kenyon-Peres to self-affine carpets"]},"model":"grok-4.3","cost_usd":0.011341,"raw_usage":{"total_tokens":4921,"prompt_tokens":554,"num_sources_used":0,"completion_tokens":52,"cost_in_usd_ticks":113412000,"prompt_tokens_details":{"text_tokens":554,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4315,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":554,"tokens_out":52,"duration_ms":26671,"temperature":1.0,"reasoning_tokens":4315,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-04T00:17:12.328721+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Compute the Hausdorff dimension of the intersection for concrete random translates of two specific Bedford-McMullen carpets and check whether the numerical value matches the dimension predicted by the entropy-based formula.","supporting_citations":[],"review_version":2}