{"id":"dfb8a9b4-91b2-4e21-afa3-52a857ead45e","arxiv_id":"1908.07624","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"Every absolutely continuous horizontal curve in the Heisenberg group whose horizontal velocity is (m-1)-times L1 differentiable almost everywhere coincides with a C^m horizontal curve outside a set of small measure, and this condition is optimal.","lead":"A pair of theorems about curves in the Heisenberg group: every sufficiently regular horizontal curve can be approximated almost everywhere by a very smooth horizontal curve, and a sharp counterexample shows the regularity condition cannot be relaxed. The result is the higher-order analogue of a known approximation theorem, with a proof that the Euclidean version does not transfer.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the main theorem correctly reduces to the published horizontal Whitney extension theorem, and the counterexample is internally consistent.","rationale":"The reader's verdict ACCEPT is justified. The positive theorem is a clean reduction to a published Whitney extension theorem, and the hypotheses are verified in detail; the one compressed step, the selection of K, is standard measure theory and does not affect correctness. The counterexample is carefully constructed and its non-approximability proof is consistent; a possible apparent contradiction with Theorem 4.1 is resolved by noting that the counterexample's f' and g' are only approximately differentiable, not L1 differentiable a.e., so the positive theorem does not apply. No internal inconsistency or missing proof was found that would change the verdict. The paper may be accepted as is, with the minor expository request to expand the K-selection argument if desired.","tokens_in":19030,"tokens_out":40112,"duration_ms":364483,"concrete_test":"Write out the omitted measure-theoretic selection of K in Theorem 4.1 explicitly: for each epsilon>0, define E_N(epsilon) as the set of points a where, for all rational r<1/N, the symmetric average of |f'-P^{m-1}_{f',a}| over B(a,r) is at most epsilon r^{m-1}, and similarly for g'; show these sets are measurable, cover almost every point, and that their intersection with the Whitney-field compact sets from Proposition 3.4 (for f, g, h) still has large measure. If such a K can be constructed with both properties, the proof of Theorem 4.1 is complete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern was identified. The central claim of Theorem 4.1 is established by reducing to the C^m horizontal Whitney extension theorem (Theorem 2.15), a published external result, and the reduction verifies all three hypotheses of that theorem. The most terse point is the simultaneous choice of the compact set K satisfying the Whitney-field condition and the uniform L1-differentiability condition (4.1); the paper defers this to 'elementary measure theory'. This is standard and fillable: f', g' are m-1 times L1 differentiable a.e., so for each epsilon one can take compact subsets of the full-measure sets where the averaged Taylor error over balls is uniformly controlled, intersect them with the Whitney-field compact sets obtained from Proposition 3.4, and then pass to a subcompact set of large measure. The counterexample in Theorem 5.1 is consistent with Theorem 4.1: although f', g' are piecewise constant, their jumps are dense and accumulate so that at typical points of (0,1)\\(I∪A) the averaged derivative fails to be o(rho), so f' and g' are not 1-L1 differentiable a.e.; hence Theorem 4.1 does not apply to the counterexample. The external Whitney theorem is a dependency, but it is prior published work, so the circularity burden is low.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a C^m Lusin approximation theorem for horizontal curves in the first Heisenberg group. The main positive result, Theorem 4.1, states that if Γ=(f,g,h):I→H is an absolutely continuous horizontal curve and f′,g′ are (m−1)-times L1 differentiable at almost every point, then Γ has the m-Lusin property: it agrees, together with its Taylor coefficients up to order m, with a C^m horizontal curve except on a set of arbitrarily small measure. The proof uses Lemma 3.1 and Lemma 3.2 to upgrade L1 differentiability of the velocity components to m-times L1 differentiability of f,g,h, Proposition 3.4 to produce Whitney fields on large compact sets, and then verifies the three hypotheses of the horizontal Whitney extension theorem (Theorem 2.15, quoted from Pinamonti–Speight–Zimmerman). The second main result, Theorem 5.1, constructs an explicit absolutely continuous horizontal curve for which f,g,h are twice L^p differentiable almost everywhere and f′,g′,h′ are once approximately differentiable almost everywhere, yet the curve does not admit a C^2 horizontal Lusin approximation. This counterexample shows that the L1 differentiability hypothesis on the velocity in Theorem 4.1 cannot be weakened to approximate differentiability, in contrast to the Euclidean situation.","tokens_in":19252,"tokens_out":7729,"duration_ms":78878,"significance":"If the results are correct, they give a natural and apparently optimal higher-order version of Lusin approximation for horizontal curves in the Heisenberg group, and the counterexample cleanly demonstrates a genuine difference between the Heisenberg and Euclidean settings. The proof of the positive theorem is well structured: the reduction to the published horizontal Whitney extension theorem is explicit, the verification of all three hypotheses of that theorem is carried out in detail, and the main estimate in the verification of hypothesis (3) is careful and quantitative. The counterexample is explicit and the obstruction is concrete, based on a density-point argument. The paper is readable and the internal logic is coherent; the main external input, Theorem 2.15, is prior published work and is used transparently as a black box, so I do not regard the dependency as circular.","major_comments":[],"minor_comments":[{"comment":"The text contains several spelling and typesetting errors, such as 'positon' in the introduction and 'APPROXIMA TION' and 'HORIZONT AL' in the running title; these should be corrected before publication.","section":"Throughout"},{"comment":"The simultaneous choice of the compact set K satisfying the Whitney-field condition, the uniform estimate (4.1), and the measure bound L1(I\\K)<η is compressed into the phrase 'elementary measure theory'; since this is the one step where three conditions are obtained together, I suggest adding a short explanation, for instance by taking compact subsets of the almost-everywhere L1-differentiability sets with uniform averaged Taylor error, intersecting them with the compact sets from Proposition 3.4, and then passing to a further compact subset of large measure.","section":"Section 4, proof of Theorem 4.1"},{"comment":"In the verification of condition (2), the passage from the identity for (TH)′ to the identities for (TH)^k would be easier to follow if the polynomials S_a and S_a^k were defined explicitly; in particular, stating that S_a^k is the k-th derivative contribution of the error term would clarify the divisibility argument and the conclusion S_a^k(a)=0.","section":"Section 4, verification of Theorem 2.15(2)"},{"comment":"Since the proof of the paper depends essentially on the horizontal Whitney extension theorem from [24], a sentence in Section 2 explicitly noting that this theorem is used as a black box and not reproved here would help orient readers who expect the paper to be fully self-contained.","section":"Section 2, Theorem 2.15"}],"recommendation":"minor_revision","confidential_remarks":"The paper relies on the authors' own previously published horizontal Whitney extension theorem [24], which includes two of the current authors as co-authors. Because that theorem is published, peer-reviewed, and used explicitly, I do not see this as a circularity or a novelty concern. The remaining suggested changes are purely expository."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper settles the natural higher-order Lusin approximation question for horizontal curves in the first Heisenberg group, and the counterexample in Section 5 is the right kind of sharpness result. I think it should go to referees.\n\nThe positive theorem (Theorem 4.1) is exactly what you'd hope: if the horizontal velocity components f', g' are (m−1)-times L1 differentiable a.e., then the absolutely continuous horizontal curve agrees with a C^m horizontal curve off a set of arbitrarily small measure. The proof is a reduction to the authors' earlier C^m Whitney extension theorem for horizontal curves (Theorem 2.15, published in Trans. AMS 2019). The three hypotheses are checked carefully: Proposition 3.4 builds the Whitney field, Lemma 3.2 gives the jet compatibility condition, and the area-discrepancy estimate in Section 4 follows from uniform L1 differentiability of f' and g' on a large compact set. This is a clean argument.\n\nSection 5 constructs an explicit horizontal curve where f,g,h are twice L^p differentiable a.e. for every p≥1 and f',g',h' are once approximately differentiable a.e., but the curve has no C^2 horizontal Lusin approximation. The construction uses dyadic intervals with small spikes, and the proof that no smooth approximation exists is a nice density argument: if a C^2 horizontal curve agreed with Γ on a large set, then two nearby density points straddling a spike would force H(y)−H(x) to be both large (from the spike height) and small (from the C^2 bound), a contradiction. The parameters are chosen to satisfy the required summability, and I did not find a gap.\n\nSoft spots are minor. The choice of the compact set K in Theorem 4.1 that simultaneously gives Whitney-field control and uniform L1 differentiability is dismissed as 'elementary measure theory'; it is standard (take compact subsets of the full-measure uniform-differentiability sets and intersect with the Whitney-field compact sets), but the paper should spell it out. Lemma 5.9 is terse: from F vanishing on a sequence of points accumulating at x it concludes F'(x)=F''(x)=0; this is correct via Taylor expansion at x, but a sentence would help. The reliance on the authors' own Whitney theorem is the main external input; since that theorem is prior published work, this is not circular, and the burden is appropriately low.\n\nWho should read this: people working on Lusin properties, Whitney extension problems, or approximation of horizontal curves in Carnot groups. It is a solid contribution that deserves a serious referee. I would recommend accept after minor revisions, mainly to expand the two terse measure-theoretic passages.","headline":"A correct and sharp higher-order Lusin approximation theorem for horizontal curves in H^1, with the main load carried by the authors' earlier Whitney extension theorem; deserves refereeing.","tokens_in":19864,"tokens_out":6680,"would_cite":true,"duration_ms":561719,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C17","26A24","28A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every absolutely continuous horizontal curve in the Heisenberg group whose velocity is $m-1$ times $L^1$-differentiable almost everywhere coincides with a $C^m$ horizontal curve except on a set of arbitrarily small measure.","keywords":["Heisenberg group","horizontal curves","Lusin approximation","L^1 differentiability","approximate differentiability","extension theorem for jets","Carnot groups","sub-Riemannian geometry"],"falsifier":"Produce a horizontal curve $\\Gamma=(f,g,h)$ satisfying the hypotheses of Theorem 4.1 for which every $C^m$ horizontal curve differs from $\\Gamma$ on a set of positive measure; equivalently, on the compact set $K$ built in the proof, show that the vertical increment $h(b)-h(a)$ cannot be reproduced from the Taylor polynomials of $f$ and $g$ up to the required order on a positive fraction of pairs $(a,b)$. Either would refute the theorem.","tokens_in":18778,"feed_emoji":"📐","tokens_out":12246,"duration_ms":110190,"temperature":0.7,"pith_summary":"The paper proves a $C^m$ Lusin approximation theorem for horizontal curves in the first Heisenberg group $\\mathbb{H}^1$. It shows that every absolutely continuous horizontal curve $\\Gamma=(f,g,h)$ for which the horizontal velocity components $f'$ and $g'$ are $m-1$ times $L^1$ differentiable almost everywhere has the $m$-Lusin property: for every $\\eta>0$ there is a $C^m$ horizontal curve that agrees with $\\Gamma$ except on a set of measure less than $\\eta$. The argument works by extracting a large compact set on which the jets of $f,g,h$ satisfy the hypotheses of the $C^m$ extension theorem for horizontal curves, then extending. The paper also constructs a horizontal curve with $f,g,h$ twice $L^p$ differentiable for every $p\\ge1$ and $f',g',h'$ once approximately differentiable that nevertheless has no $C^2$ horizontal Lusin approximation, showing the hypothesis cannot be weakened to approximate differentiability. The result matters because it identifies the regularity of the velocity, not of the curve itself, that controls smoothing in sub-Riemannian geometry.","feed_headline":"Rough horizontal curves in the Heisenberg group get C^m smoothings","feed_subtitle":"When the horizontal velocity is m-1 times L^1-differentiable, the curve matches a C^m horizontal curve except on tiny sets.","key_machinery":"The carrying object is the extension theorem for $C^m$ horizontal curves in $\\mathbb{H}^1$ (Theorem 2.15), quoted from an earlier paper. It says a jet $(F,G,H)$ of order $m$ on a compact set $K$ extends to a $C^m$ horizontal curve exactly when three conditions hold: the jet is a $C^m$ Whitney field (Taylor remainders are $o(|a-b|^{m-k})$); the derivatives satisfy the Leibniz-type relation $H^k = 2\\sum_{i=0}^{k-1}\\binom{k-1}{i}(F^{k-i}G^i - G^{k-i}F^i)$; and an area discrepancy $A(a,b)/V(a,b)\\to 0$ uniformly, where $A$ measures the mismatch between the vertical increment $h(b)-h(a)$ and the increment predicted by the Taylor polynomials of $f$ and $g$, while $V$ controls the scale. The proof of Theorem 4.1 chooses a large compact set $K$ using Lemma 3.1 and Proposition 3.4 (integration of $L^1$ derivatives and a polynomial estimate on density sets), then verifies the three hypotheses, with the $L^1$ differentiability of $f',g'$ giving the uniform estimates needed for the area discrepancy.","core_discovery":"On the paper's own terms, the central discovery is Theorem 4.1: if $\\Gamma=(f,g,h):I\\to\\mathbb{H}$ is an absolutely continuous horizontal curve and $f',g'$ are $m-1$ times $L^1$ differentiable at almost every point of $I$, then $\\Gamma$ has the $m$-Lusin property; moreover the approximating $C^m$ horizontal curve can be chosen so that its derivatives up to order $m$ agree with the jets of $\\Gamma$ on the large set. The converse half, Theorem 5.1, constructs an explicit absolutely continuous horizontal curve for which $f,g,h$ are twice $L^p$ differentiable for all $p\\ge1$ and $f',g',h'$ are once approximately differentiable almost everywhere, yet no $C^2$ horizontal curve agrees with it on a set of positive measure. Together these show that the $L^1$ differentiability of the velocity is the right hypothesis and that the Euclidean sufficient condition, approximate differentiability of the curve itself, is not sufficient in the Heisenberg group, because the approximating curve must remain horizontal.","pith_inferences":["Because the positive proof reduces to verifying the three hypotheses of the extension theorem on a large compact set, any Carnot group with a $C^m$ extension theorem for horizontal curves will inherit the same $m$-Lusin property for curves with $L^1$-differentiable velocity; the group-specific work is concentrated in the extension theorem.","The counterexample's mechanism, vertical increments that concentrate on small intervals and cannot be matched by any horizontal $C^2$ jet, suggests that the area discrepancy is the right quantitative obstruction, and one could try to build counterexamples in other Carnot groups by engineering similar concentration.","A testable open direction is whether the theorem survives if $L^1$ differentiability of $f',g'$ is replaced by $L^p$ differentiability for some fixed $p>1$; the proof's estimates are written for $L^1$ averages, so this is not a direct corollary.","If the quoted $C^m$ extension theorem is available in higher-dimensional Heisenberg groups $\\mathbb{H}^n$, the same compact-set construction should give the analogue of Theorem 4.1; the authors expect this, but the paper proves only the first Heisenberg group."],"forward_implications":["Every horizontal curve in $\\mathbb{H}^1$ with $(m-1)$-fold $L^1$-differentiable velocity admits a $C^m$ horizontal approximation, so the $C^1$ result known for all absolutely continuous horizontal curves extends to all orders $m$.","The approximating $C^m$ curve can be arranged to match the original curve and all its derivatives up to order $m$ on the large agreement set.","Approximate differentiability of the curve itself, which is sufficient in Euclidean space, is not sufficient in $\\mathbb{H}^1$: the paper's counterexample has $f,g,h$ twice $L^p$-differentiable for every $p\\ge1$ and $f',g',h'$ once approximately differentiable, yet no $C^2$ horizontal Lusin approximation exists.","The $L^1$ differentiability hypothesis on the velocity is therefore the correct threshold in the Heisenberg group, at least for $m=2$."],"supporting_citations":[{"why":"Provides the $C^m$ extension theorem for horizontal curves in the Heisenberg group, the external tool whose three hypotheses the proof verifies.","marker":"[24]"},{"why":"Supplies the Euclidean $m$-Lusin theorem and the measurability of approximate derivatives; the paper adapts its compact-set strategy.","marker":"[20]"},{"why":"Gives the polynomial estimate used (as the De Giorgi lemma) to convert approximate differentiability on large sets into a jet field.","marker":"[5]"},{"why":"The classical extension theorem for $C^m$ jets, which supplies the jet formalism and the Whitney-field condition.","marker":"[31]"},{"why":"Establishes the $C^1$ horizontal Lusin property in Heisenberg groups and shows failure in the Engel group, providing the base case and motivation.","marker":"[28]"},{"why":"Proves the earlier $C^1$ extension theorem for horizontal curves in the Heisenberg group, a direct predecessor of the $C^m$ theorem.","marker":"[32]"}],"fun_headline_variants":["Heisenberg smoothing: L^1-differentiable velocity is enough","Optimal Lusin theorem for Heisenberg horizontal curves","Approximate differentiability fails for Heisenberg smoothing","Heisenberg smoothing needs L^1 velocity, not approximate","C^m smoothings for Heisenberg curves: L^1 velocity optimal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes, without proving it here, the quoted $C^m$ extension theorem for horizontal curves in the Heisenberg group (Theorem 2.15); if that theorem were false or incomplete, the main positive result would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Heisenberg smoothing: L^1-differentiable velocity is enough","Optimal Lusin theorem for Heisenberg horizontal curves","Approximate differentiability fails for Heisenberg smoothing","Heisenberg smoothing needs L^1 velocity, not approximate","C^m smoothings for Heisenberg curves: L^1 velocity optimal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001177,"raw_usage":{"total_tokens":4831,"prompt_tokens":877,"completion_tokens":3954,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":3870}},"tokens_in":493,"tokens_out":3954,"duration_ms":29552,"temperature":1.0,"reasoning_tokens":3870,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:01:41.859298+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce a horizontal curve $\\Gamma=(f,g,h)$ satisfying the hypotheses of Theorem 4.1 for which every $C^m$ horizontal curve differs from $\\Gamma$ on a set of positive measure; equivalently, on the compact set $K$ built in the proof, show that the vertical increment $h(b)-h(a)$ cannot be reproduced from the Taylor polynomials of $f$ and $g$ up to the required order on a positive fraction of pairs $(a,b)$. Either would refute the theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the $C^m$ extension theorem for horizontal curves in the Heisenberg group, the external tool whose three hypotheses the proof verifies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Euclidean $m$-Lusin theorem and the measurability of approximate derivatives; the paper adapts its compact-set strategy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the polynomial estimate used (as the De Giorgi lemma) to convert approximate differentiability on large sets into a jet field."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classical extension theorem for $C^m$ jets, which supplies the jet formalism and the Whitney-field condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the $C^1$ horizontal Lusin property in Heisenberg groups and shows failure in the Engel group, providing the base case and motivation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the earlier $C^1$ extension theorem for horizontal curves in the Heisenberg group, a direct predecessor of the $C^m$ theorem."}],"review_version":1}