{"id":"25f6b29b-3c4e-4909-a865-dbc705f2f6ff","arxiv_id":"2607.15635","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":5,"one_line_summary":"A spectral-gap criterion gives Sobolev regularity for prescribed projections of self-similar measures and yields explicit examples such as singular measures with all line projections smooth.","lead":"This paper proves a concrete condition that decides when one individual projection of a self-similar fractal measure has a smooth density, based on how fast the rotations mix and on a projection-relative dimension. Using explicit Ramanujan rotation sets, it builds surprising examples, including singular measures whose every line projection is absolutely continuous.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"LPS equality (1.3) is likely false (e.g., p=5), but the applications only need the upper bound; replace '=' by '≤' and the proofs survive.","rationale":"The reader correctly identified the LPS spectral bound as the delicate external input for the explicit applications, but framed the danger as 'replacing equality by a bound with slack in the wrong direction would break them.' In fact the LPS theorem provides an upper bound, and the wrong-direction slack would be an actual norm larger than 2√p/(p+1), which would contradict that theorem. The real issue is that (1.3) states an exact equality that is already false for p=5, a prime used in the paper. However, every subsequent use of (1.3) is conservative: a smaller true norm gives a larger spectral gap and a stronger spherical-average contraction, so Corollaries 1.3–1.7 remain valid if '=' is replaced by '≤'. The central criterion, Theorem 1.9, is independent of LPS and its proof is internally consistent. Thus the paper needs a correction rather than rejection; the verdict should be conditional on correcting (1.3) and the corresponding '=' statements in §5 to upper bounds.","tokens_in":47341,"tokens_out":58679,"duration_ms":643491,"concrete_test":"For the p=5 Ramanujan set (six rotations with cos θ = −3/5), compute the spin-ℓ eigenvalues χ_ℓ(θ)/(2ℓ+1) for ℓ = 1,...,20 and take the supremum. If it is strictly less than √5/3, then (1.3) is false as an equality. Then verify that every theorem in §5 that quotes (1.3) only uses the inequality ‖P‖ ≤ 2√p/(p+1), by rechecking the displayed numerical inequalities with the true (smaller) norm; if they still pass, the applications are unaffected.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Equation (1.3) asserts that for every prime p ≡ 1 mod 4 the normalized LPS averaging operator on L^2_0(SO(3)) has norm exactly 2√p/(p+1). This is not the right reading of the LPS theorem: the Ramanujan bound is an upper bound, and it is not attained for each p. For the p=5 set actually used in Corollaries 1.5–1.7, the six rotations come from quaternions of norm 5, all with rotation angle θ satisfying cos θ = −3/5. On the spin-ℓ representation of SO(3), the averaging operator is scalar with eigenvalue χ_ℓ(θ)/(2ℓ+1), where χ_ℓ is the character. Computing χ_1 = −1/5, χ_2 = −19/25, χ_3 = 139/125, etc., gives sup_ℓ |χ_ℓ(θ)/(2ℓ+1)| ≈ 0.159, far below √5/3 ≈ 0.745. So (1.3) is not an equality for p=5. This is still not fatal: every numerical use of (1.3) is in the direction where replacing the equality by the true upper bound only makes the spectral gap larger (i.e., −log‖P‖ larger). Corollaries 1.3–1.7 can be re-proved verbatim with '≤' in place of '='. The manuscript should state and use the upper bound, not the false equality.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a Furstenberg-type criterion for individual orthogonal projections of self-similar measures to be absolutely continuous with quantified Sobolev/Besov regularity. The main result, Theorem 1.9, says that if the rotational part of the IFS is exponentially mixing (spectral gap) on the orbit of the prescribed subspace V, and if the rate of mixing is fast compared with a certain orbit-relative L^q dimension of the measure, then the projected measure is absolutely continuous and its density lies in a Besov space. The proof combines a model decomposition for non-uniform contraction ratios, random-walk estimates on the group/orbit, Sobolev embedding, and Littlewood–Paley analysis. The abstract applications use Lubotzky–Phillips–Sarnak Ramanujan sets in SO(3): singular self-similar measures with every line projection absolutely continuous, measures with arbitrarily small Fourier dimension but smooth projections in all but one explicit direction, and a self-similar Salem measure with C^2 density. The paper is carefully written and the main mechanism is original, but the explicit applications rest on a false reading of the LPS theorem that needs correction.","tokens_in":47757,"tokens_out":9286,"duration_ms":103881,"significance":"If the results hold, this is the first general fully explicit criterion ensuring smoothness of a prescribed projection of a self-similar measure, and the applications are striking: they show that spectral gap can completely overcome Marstrand-type exceptional directions, even when the ambient measure is singular and has very small Fourier dimension. The paper also provides a clean Besov-space framework and an orbit-relative dimension that may be useful beyond this setting. The proofs are detailed and largely self-contained, and the numerical conditions are concrete and checkable. The main reservation concerns not the strategy but the incorrect use of the LPS spectral equality in the explicit applications; this is fixable and the applications survive because the actual direction of the inequality is favorable.","major_comments":[{"comment":"The statement that for every prime p≡1 mod 4 the LPS Ramanujan set satisfies ∥P_G∥_{L^2_0(SO(3))}=2√p/(p+1) is not correct. The LPS theorem gives an upper bound, and the bound need not be attained for each p. For p=5, the six rotations from quaternions of norm 5 all have rotation angle θ with cos θ=−3/5; on the spin-ℓ representation the averaging operator is scalar with eigenvalue χ_ℓ(θ)/(2ℓ+1), and the supremum over ℓ is ≈0.159, far below √5/3≈0.745. This is not a harmless citation slip: Sections 5.1, 5.2, 5.3 and 5.6 use Eq. (1.3) as an equality in the numerical verifications. In every use the inequality direction is favorable — replacing '=' by '≤' makes the spectral gap larger — so the proofs survive verbatim. Nevertheless, the manuscript must be corrected to state the LPS result as an upper bound, and the wording 'precisely'/'achieved' must be adjusted. Because the explicit applicat","section":"§1.3, Eq. (1.3)"}],"minor_comments":[{"comment":"The assertion dim_F μ_{r,ε} ≤ dim_F λ_ε is described as 'direct to check'. It is true, since the Fourier transform of the fourth-coordinate marginal equals the restriction of the ambient Fourier transform to the vertical frequency axis, but a one-sentence justification would help the reader, especially because this is used to conclude that the ambient Fourier dimension is small.","section":"§5.3, Corollary 1.5"},{"comment":"In the numerical check, the expression 'log 18/(2√17)' is ambiguous. It should be written as −log(2√17/18) or with explicit parentheses.","section":"§5.1"},{"comment":"The notation R_p is used both for the set of rotations and for the averaging operator norm, and the dependence on p is sometimes omitted. Adding a subscript or explicit dependence would improve readability.","section":"§1.3 and §5.2"},{"comment":"The set in the displayed inequality is written as 'sup (t0u ∪ {...})'; the braces are visually confusing. This is a minor typographical issue.","section":"§3.2, Lemma 3.4"}],"recommendation":"major_revision","confidential_remarks":"The only substantive technical concern is the false equality (1.3). It is not a reason to reject: replacing the equality by the correct upper bound only strengthens the numerical inequalities that drive the applications. I recommend major revision because the authors must correct the statement and re-verify the explicit computations under the correct reading. Once that is done, the paper is likely acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing to know: this paper delivers what it promises. Theorem 1.9 gives a general, explicit spectral-gap criterion for a prescribed projection of a self-similar measure to be absolutely continuous with quantified Besov regularity, and the applications are genuinely striking: singular self-similar measures with every line projection absolutely continuous, measures of arbitrarily small Fourier dimension with smooth projections in all but an explicit exceptional direction, and a self-similar Salem measure with a C^2 density. The orbit-relative L^q dimension is a sensible and apparently new quantity, and the proof machinery (model tree decomposition, Sobolev estimates on orbits, Littlewood-Paley argument) is detailed and coherent. I spot-checked several numerical inequalities and they hold. The comparison with prior work is honest and accurate: Lindenstrauss–Varjú and Kittle–Kogler concern the ambient measure, not projections, and their constants are non-explicit; Rapaport's example shows density of rotations is insufficient, so the spectral-gap assumption is well motivated.\n\nThe one real soft spot is equation (1.3). As the stress-test note shows, the LPS theorem gives an upper bound, not an equality attained for every p; for p=5 the actual norm on L^2_0(SO(3)) is far below 2√5/6. This should be fixed in the manuscript. That said, it is a minor fix: every numerical use of (1.3) is in the direction where replacing '=' by '≤' only makes the spectral gap larger, so Corollaries 1.3–1.7 survive verbatim. I would not treat this as a load-bearing flaw.\n\nThe other soft spots are the expected “direct to check” claims, e.g., dim_F μ ≤ dim_F λ_ε in Corollary 1.5, and some uniform estimates in Section 5. They are plausible and do not affect the central argument. I also find the entropy-dimension comparison used to deduce Theorem 1.2 from Theorem 1.9 natural, though it relies on prior results (Feng–Hu, Shmerkin–Solomyak) rather than being reproved here.\n\nBottom line: this is a strong paper for fractal geometry and dynamical systems readers. It deserves a serious referee and publication after the LPS-equality correction. I would cite it and bring it to a reading group.","headline":"First fully explicit criterion for smoothness of prescribed projections of self-similar measures, with striking constructions from Ramanujan sets; the core proofs are sound, though the LPS equality (1.3) should be weakened to the upper bound it actually supports.","tokens_in":48221,"tokens_out":1030,"would_cite":true,"duration_ms":14746,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28A80","42B10","37C45"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, in dimensions at least three, a prescribed projection of a self-similar measure is absolutely continuous with a Besov-class density whenever the rotations mix exponentially fast compared with the measure's orbit-rela","keywords":["self-similar measures","orthogonal projections","absolute continuity","spectral gap","Fourier dimension","Besov spaces","spherical averages","Ramanujan sets"],"falsifier":"Compute the operator norm on L^2_0(SO(3)) of the averaging operator over the explicit 18-element rotation set used in the R^3 example; if it exceeds 2√17/18, then the claimed absolute continuity of all line projections is not certified by the criterion, and if some line projection is actually singular, the criterion itself is false. More generally, for the R^4 family, evaluate both sides of the spectral inequality for a chosen prime p and dimension D; an inequality that holds while the projected measure fails to be in L^2 would refute the theorem.","tokens_in":47236,"feed_emoji":"📐","tokens_out":9288,"duration_ms":96680,"temperature":0.7,"pith_summary":"Self-similar measures in R^d, d≥3, can be certified projection by projection: the paper proves that a given orthogonal projection is absolutely continuous, with a quantified Besov regularity of its density, as soon as the rotational part of the iterated function system mixes exponentially fast at a rate that beats the measure's dimension relative to that projection's orbit. The theorem reduces this to an inequality between the spectral gap of an averaging operator on the orbit and a relative L^q dimension, and the proof is Fourier-analytic: it decomposes the projected Fourier transform into dyadic annuli, runs the self-similarity through a random walk on rotated directions, and controls the error by Sobolev embedding on the orbit. Using explicitly constructed rotation sets with optimal spectral gap, the paper produces singular self-similar measures whose every line projection is absolutely continuous, measures of arbitrarily small Fourier dimension whose projections are smooth in all but a fully explicit exceptional set of directions, and a non-trivial self-similar measure that is Salem (Fourier dimension equals Hausdorff dimension) with a C^2_0 density. This matters because it is the first general and fully explicit criterion for smoothness of a prescribed projection, and it shows that singular ambient measures can still have remarkably smooth projection structure.","feed_headline":"When mixing beats dimension, fractal projections turn absolutely continuous","feed_subtitle":"New criterion quantifies the smoothness of the projected density from the rotation spectrum, with explicit examples in R^3 and R^4.","key_machinery":"The central object is the averaging operator P_{a,V} on the compact homogeneous orbit O_V of the prescribed subspace under the rotation group; its norm on the zero-mean L^2 space measures how fast rotational mixing erases the non-invariant part of any function on the orbit. The comparison quantity is the orbit-relative L^q dimension dim^V_q ν, defined by summability of orbit-averaged Littlewood-Paley norms of the projected measure; it is the integrability threshold of the spherical averages that the proof tries to dominate. The spectral inequality is the condition that the exponential mixing term decays faster than the polynomial growth of the annuli and of the Sobolev norm of the frequency-","core_discovery":"The central claim (Theorem 1.9) is a conditional smoothness criterion. For a self-similar IFS on R^d (d≥3) with measure ν, fix a k-plane V and let G be the closed rotation group with orbit O_V=G·V. For each contraction modulus a, let P_{a,V} be the averaging operator on the zero-mean L^2 space of O_V, and let dim^V_q ν be the L^q dimension of ν relative to O_V. If the spectral ratio Λ = [−log Σ_a r_{p_a}‖P_{a,V}‖] / max_i(−log|r_i|) exceeds (b(q−1)+b(σ+b(q−1)))/(S−b) for some k<b<S<dim^V_q ν (with σ>½ dim O_V and, when q<2, σ<q), then π_V ν is absolutely continuous and its density lies in the Besov space B^{(b−k)/q′}_{q,q}(V), hence in L^q(V).","pith_inferences":["The spectral inequality probably admits a sharper form in which the threshold depends on the whole Lyapunov spectrum; replacing the maximum contraction rate by the Lyapunov exponent, as the paper's remark does, suggests that generically the threshold is easier to satisfy, so one may expect many more explicit examples than the ones constructed.","A natural test is to push the construction toward the boundary: letting the dimension D approach 4 in the R^4 example should make the L^2 density approach C^1 regularity, indicating a critical threshold at which the density gains a full derivative.","The exceptional-direction mechanism in the small-Fourier-dimension example — a singular marginal along one coordinate forcing singularity of any projection containing that direction — suggests a general slicing principle: for product-like self-similar couplings, the set of singular projection directions is controlled by the most singular marginal, not by the full measure.","Because the criterion is fully explicit, it can be used as a computational certificate: for a given algebraic IFS one can numerically bound the operator norms and relative dimensions and verify or refute absolute continuity of a specific direction before any symbolic integration."],"forward_implications":["For any self-similar measure satisfying the spectral inequality, every prescribed projection is not merely dimension-preserving but has an L^q density with explicit Besov regularity; the certificate is a computation of an operator norm and a relative dimension.","Singular self-similar measures on R^3 exist whose every line projection is absolutely continuous, so singularity of the ambient measure is no obstruction to smoothness of all its line projections.","On R^4, for any dimension D between (3+√21)/2 and 4, there is a singular self-similar measure of dimension D all of whose line projections have L^2 densities.","The phenomenon is independent of Fourier dimension: for any η>0 one can force the Fourier dimension below η while keeping all projections onto planes not containing a fixed direction smooth up to Sobolev order (4−k)/2−τ.","There exist non-trivial self-similar measures on R^3 that are Salem (Fourier dimension equals Hausdorff dimension) and have a C^2_0 density with non-empty interior of the support."],"fun_headline_variants":["Mixing rate beats dimension: smooth projections for self-similar measures","Fractal projections absolutely continuous when mixing outpaces dimension","New criterion: rotation mixing speed determines smoothness of fractal projections","Explicit singular fractals with smooth projections in every direction"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"For the explicit applications, everything rests on the exact spectral norm of the averaging operator on the optimally mixing rotation set: if that norm were even slightly larger than the stated value, the numerical inequalities that drive the examples would no longer hold.","fun_headline_variants_meta":{"raw":{"variants":["Mixing rate beats dimension: smooth projections for self-similar measures","Fractal projections absolutely continuous when mixing outpaces dimension","New criterion: rotation mixing speed determines smoothness of fractal projections","Explicit singular fractals with smooth projections in every direction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000324,"raw_usage":{"total_tokens":1653,"prompt_tokens":742,"completion_tokens":911,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":841}},"tokens_in":486,"tokens_out":911,"duration_ms":9952,"temperature":1.0,"reasoning_tokens":841,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T22:43:57.468917+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the operator norm on L^2_0(SO(3)) of the averaging operator over the explicit 18-element rotation set used in the R^3 example; if it exceeds 2√17/18, then the claimed absolute continuity of all line projections is not certified by the criterion, and if some line projection is actually singular, the criterion itself is false. More generally, for the R^4 family, evaluate both sides of the spectral inequality for a chosen prime p and dimension D; an inequality that holds while the projected measure fails to be in L^2 would refute the theorem.","supporting_citations":[],"review_version":1}