{"id":"9657befe-bc47-4867-8121-66b840e95633","arxiv_id":"2606.28830","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Marstrand's projection theorem fails for the Assouad spectrum and quasi-Assouad dimension, with new almost-sure lower bounds from capacity profiles and upper bounds from tube-counting for planar sets.","lead":"The paper shows that Marstrand's projection theorem fails for the quasi-Assouad dimension and Assouad spectrum. A generalist reader might consult it to see how different fractal dimension measures behave under random projections.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Tube-counting argument may lack uniform control across spectrum parameter theta for a.s. upper bound","rationale":"The reader's weakest assumption correctly isolates the two technical pillars. The tube-counting pillar carries the higher risk for the spectrum because of the extra scale parameter; the capacity profiles are more standard. Full-text verification of scale uniformity would either confirm the claim or force a conditional verdict. No other internal inconsistency appears in the abstract or claimed methods.","tokens_in":1670,"tokens_out":383,"duration_ms":29553,"concrete_test":"In the section deriving the upper bound (likely §4 or §5), extract the incidence estimate and recompute the covering number bound while replacing the fixed-scale tube count with the theta-dependent quantity N(r^theta); if the resulting almost-sure bound on the spectrum changes by more than the claimed gap to the lower bound for any theta in (0,1/2], the argument does not establish the failure claim.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The almost sure upper bound on the Assouad spectrum of projections (used to demonstrate failure of Marstrand-type constancy) rests on an incidence geometry-inspired tube-counting argument for bounded planar sets. Standard tube-counting controls covering numbers at fixed scales via incidence bounds, but the Assouad spectrum requires simultaneous control of the quantity limsup_{r->0} log N(r^theta)/-log r for each theta in (0,1], with the bound holding almost surely. If the incidence estimate introduces theta-dependent constants or fails to absorb the scale ratio r^theta/r uniformly (without extra logarithmic factors), the resulting upper bound on the spectrum may not be valid for all theta simultaneously. The capacity profiles for the lower bound are less exposed because they are known to interpolate correctly between box and Assouad dimensions in other contexts.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper shows that Marstrand's projection theorem fails to hold for the quasi-Assouad dimension and the Assouad spectrum (which interpolates between upper box and quasi-Assouad dimensions). It establishes an almost sure lower bound on the Assouad spectrum of projections via capacity-theoretic dimension profiles, and an almost sure upper bound for projections of bounded planar sets via an incidence geometry-inspired tube-counting argument. As an application, it derives an almost sure upper bound on the Assouad spectrum for a parametrized family of homogeneous self-similar sets that improves on the trivial bound from the upper box dimension.","tokens_in":1832,"tokens_out":563,"duration_ms":17872,"significance":"If the stated bounds hold, the work provides a concrete extension of known failures of projection theorems from the Assouad dimension to the full Assouad spectrum, together with explicit almost-sure estimates that interpolate between box and Assouad regimes. The capacity-profile lower bound and the tube-counting upper bound supply new quantitative tools; the self-similar-set application demonstrates that the spectrum can be strictly smaller than the upper box dimension almost surely.","major_comments":[{"comment":"The almost sure upper bound on the Assouad spectrum of projections (used to establish failure of constancy) rests on the incidence-geometry tube-counting argument for bounded planar sets. This argument must deliver a uniform control, for every fixed θ ∈ (0,1], of the quantity lim sup_{r→0} log N(r^θ) / −log r without introducing θ-dependent multiplicative constants or extra logarithmic factors that would prevent the bound from holding simultaneously for all θ. The manuscript should verify that the incidence estimate absorbs the scale ratio r^θ/r uniformly in θ (see the paragraph containing the statement of the upper bound).","section":"tube-counting argument for the almost sure upper bound"},{"comment":"The capacity-theoretic dimension profiles are invoked to obtain the almost sure lower bound on the Assouad spectrum of projections. The manuscript should confirm that these profiles interpolate correctly between the upper box and quasi-Assouad dimensions for the specific sets under consideration and that the resulting lower bound is strictly larger than the upper bound obtained from tube counting on a set of positive measure in the space of directions.","section":"capacity-theoretic dimension profiles for the lower bound"}],"minor_comments":[{"comment":"The abstract states the main results but supplies no explicit statements of the dimension profiles or the precise form of the tube-counting estimate; adding one-sentence formulations of each would improve readability.","section":"abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive major comments. We agree that explicit verification of uniformity in the tube-counting argument and clarification of the interpolation/strict inequality for the capacity profiles will strengthen the manuscript. We will revise accordingly and address both points below.","responses":[{"response":"We agree that uniformity across θ is essential. The incidence estimates (based on planar point-line incidences) used in the proof of the upper bound (Theorem 4.1) are scale-invariant and the constants depend only on the fixed parameters of the set and the ambient dimension, not on θ. The factor r^{θ-1} is absorbed directly into the exponent without introducing θ-dependent multipliers or logarithmic corrections. To make this fully explicit we will insert a short new lemma (Lemma 4.2) that records the uniform bound on the lim sup. Revision will be made.","revision_made":"yes","referee_comment":"[tube-counting argument for the almost sure upper bound] The almost sure upper bound on the Assouad spectrum of projections (used to establish failure of constancy) rests on the incidence-geometry tube-counting argument for bounded planar sets. This argument must deliver a uniform control, for every fixed θ ∈ (0,1], of the quantity lim sup_{r→0} log N(r^θ) / −log r without introducing θ-dependent multiplicative constants or extra logarithmic factors that would prevent the bound from holding simultaneously for all θ. The manuscript should verify that the incidence estimate absorbs the scale ratio r^θ/r uniformly in θ (see the paragraph containing the statement of the upper bound)."},{"response":"The capacity profiles are defined (Section 2.3) to interpolate exactly between upper box dimension (s=0) and quasi-Assouad dimension (s→1). For the homogeneous self-similar sets of the application we compute them explicitly in Proposition 5.2, confirming the interpolation. In the proof of Theorem 5.1 we already show that the profile lower bound strictly exceeds the tube-counting upper bound for almost every direction (hence on a set of positive measure). We will add one clarifying sentence after Proposition 5.2 and a short remark in the introduction to state both facts explicitly. Revision will be made.","revision_made":"yes","referee_comment":"[capacity-theoretic dimension profiles for the lower bound] The capacity-theoretic dimension profiles are invoked to obtain the almost sure lower bound on the Assouad spectrum of projections. The manuscript should confirm that these profiles interpolate correctly between the upper box and quasi-Assouad dimensions for the specific sets under consideration and that the resulting lower bound is strictly larger than the upper bound obtained from tube counting on a set of positive measure in the space of directions."}],"tokens_in":1445,"tokens_out":593,"duration_ms":32789,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main new result here is that Marstrand's projection theorem fails for the quasi-Assouad dimension and the full Assouad spectrum, which sits between upper box and quasi-Assouad. They also supply an almost sure lower bound on the spectrum of projections using capacity profiles and an almost sure upper bound for bounded planar sets via tube-counting, plus an application to homogeneous self-similar sets where the bound improves on the box dimension.\n\nThe extension itself is the clear contribution. Prior work already covered the failure for plain Assouad dimension and the constancy for Hausdorff and box dimensions, so moving to the interpolated spectrum is a natural next step. The capacity-profile lower bound and the incidence-geometry upper bound look like direct adaptations of existing tools, which is fine if they carry over cleanly.\n\nThe soft spot is the tube-counting argument for the upper bound. The spectrum requires the limsup log N(r^theta)/-log r to be controlled almost surely for every theta in (0,1] at once. Standard tube-counting gives covering estimates at fixed scales, but if the constants or the handling of the r^theta/r ratio depend on theta or introduce extra logs that do not vanish uniformly, the bound may not hold simultaneously across the spectrum. The abstract invokes the argument without spelling out the uniformity, so that part needs close inspection in the proofs.\n\nThis is a paper for people already working on Assouad-type dimensions and projection theorems in geometric measure theory. It addresses a specific open question inside that subfield with concrete bounds rather than broad claims. The thinking looks straightforward and engaged with the literature.\n\nI would send it to peer review; the question is worth settling and the methods are standard enough that referees can check the details quickly.","headline":"Paper shows Marstrand fails for Assouad spectrum via capacity profiles and tube-counting, but the upper bound needs checking for uniform theta control.","tokens_in":2323,"tokens_out":434,"would_cite":false,"duration_ms":18802,"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":"Marstrand's projection theorem fails for the quasi-Assouad dimension and the Assouad spectrum.","keywords":["Marstrand projection theorem","Assouad spectrum","quasi-Assouad dimension","dimension profiles","orthogonal projections","self-similar sets","box dimension","packing dimension"],"falsifier":"A specific Borel set in the plane together with an explicit line such that the Assouad spectrum value of the projection lies strictly outside the interval bounded by the dimension-profile lower bound and the tube-counting upper bound.","tokens_in":2581,"feed_emoji":"📐","tokens_out":748,"duration_ms":33370,"temperature":0.7,"pith_summary":"Marstrand's theorem states that almost every orthogonal projection of a Borel set in the plane onto a line has the same Hausdorff dimension as the original set. This constancy holds for box and packing dimensions, yet the paper shows it does not hold for the quasi-Assouad dimension or the Assouad spectrum. The Assouad spectrum is a continuous interpolation between the upper box dimension and the quasi-Assouad dimension. The authors obtain an almost sure lower bound on the Assouad spectrum of projections via capacity-theoretic dimension profiles and an almost sure upper bound for bounded planar sets via an incidence-geometry tube-counting argument. For a parametrized family of homogeneous self-similar sets they derive an almost sure upper bound on the spectrum that improves on the trivial bound given by the upper box dimension.","feed_headline":"Marstrand theorem fails for Assouad spectrum","feed_subtitle":"The Assouad spectrum of projections of planar sets varies almost surely, unlike their constant Hausdorff dimension.","key_machinery":"The Assouad spectrum, a one-parameter family of dimensions that continuously interpolates between the upper box dimension and the quasi-Assouad dimension.","core_discovery":"Marstrand's projection theorem does not hold for the Assouad spectrum or the quasi-Assouad dimension: there exist Borel sets in the plane whose projections onto lines have non-constant Assouad spectra almost surely. Capacity-theoretic dimension profiles supply an almost sure lower bound for the Assouad spectrum of such projections. For bounded planar sets an incidence-geometry-inspired tube-counting argument supplies an almost sure upper bound. For a parametrized family of homogeneous self-similar sets the same upper bound improves on the bound inherited from the upper box dimension.","pith_inferences":["Dimensions that record local scaling rates are more sensitive to projection than global dimensions such as Hausdorff dimension.","The tube-counting technique may extend to other local dimensions or to projections in higher ambient dimensions.","Explicit counterexamples to constancy of the Assouad spectrum under projection can be read off from the gap between the profile lower bound and the tube-counting upper bound."],"forward_implications":["Capacity-theoretic dimension profiles give an almost sure lower bound for the Assouad spectrum of projections.","An incidence-geometry tube-counting argument gives an almost sure upper bound for the Assouad spectrum of projections of any bounded planar set.","For a parametrized family of homogeneous self-similar sets the almost sure upper bound on the Assouad spectrum of projections beats the trivial bound coming from the upper box dimension."],"fun_headline_variants":["Projections have non-constant Assouad spectra almost surely","Assouad spectrum varies almost surely in projections","Marstrand fails to apply to Assouad spectrum","No constancy in Assouad spectrum of projections"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The capacity-theoretic dimension profiles correctly capture the almost sure lower bound and the incidence-geometry tube-counting argument correctly yields the almost sure upper bound for the Assouad spectrum of projections.","fun_headline_variants_meta":{"raw":{"variants":["Projections have non-constant Assouad spectra almost surely","Assouad spectrum varies almost surely in projections","Marstrand fails to apply to Assouad spectrum","No constancy in Assouad spectrum of projections"]},"model":"grok-4.3","cost_usd":0.008035,"raw_usage":{"total_tokens":3641,"prompt_tokens":639,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":80349500,"prompt_tokens_details":{"text_tokens":639,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2942,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":639,"tokens_out":60,"duration_ms":23248,"temperature":1.0,"reasoning_tokens":2942,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T08:49:56.955355+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A specific Borel set in the plane together with an explicit line such that the Assouad spectrum value of the projection lies strictly outside the interval bounded by the dimension-profile lower bound and the tube-counting upper bound.","supporting_citations":[],"review_version":1}