{"id":"b1188fc0-5f2e-45b7-b889-2ba97bb98e18","arxiv_id":"2506.20390","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For dilation sets with bounded Assouad dimension α, the circular maximal function is proven of restricted weak type at the endpoint Q_{4,α}, and the fractal local smoothing estimate holds for an extended range of q.","lead":"This paper closes the last open endpoint estimate for the circular maximal function over fractal sets of dilations, and it widens the range of p and q for which a sharp local smoothing estimate for the wave equation is known. Endpoint bounds decide exactly when spherical averaging operators are bounded between L^p and L^q spaces, and the problem was left open in work by Roos and Seeger.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4 depends on Corollary 3.5, whose uniform-in-θ proof is only sketched ('Instead of reproducing all the details'); if the T_θ stability assertion fails, the extended local smoothing range collapses.","rationale":"The reader's weakest-assumption analysis is accurate: the only place where a load-bearing step is explicitly deferred is Corollary 3.5 in Section 3.1, and the extended local smoothing theorem depends on it. I agree that this warrants a conditional verdict. The endpoint result Theorem 1.2, which is the primary title claim, is proved in Section 2 using Proposition 2.1, Lemma 2.2, and the known bilinear cone restriction estimate; I did not find a substantive gap in that chain. Thus the concern is not about the central endpoint claim but about the advertised extension of the local smoothing range. I also note the separate α = 1 parameter issue in the proof of Proposition 3.1, but it is minor and repairable, so it does not change the verdict. The proposed concrete test—a complete uniform-in-θ verification of Lemma 3.3 and the ε-removal step—directly settles whether the concern lands. No ad hominem, no manufactured objection: the paper is serious and largely self-contained, but Theorem 1.4 currently rests on an asserted stability result rather than a demonstrated one.","tokens_in":33325,"tokens_out":23793,"duration_ms":225041,"concrete_test":"Write out the full uniform-in-θ proof of Lemma 3.3 for the operators T_θ. Concretely: fix R and 0 ≤ θ ≤ θ_0, perform the wave-packet decomposition for T_θ at scale R^{1/2}, and verify the estimates (3.10) and (3.11) with constants independent of θ. If the wave-packet coefficients or the L^2 bounds in (3.10) pick up a factor diverging as θ → 0, or if the geometric estimate (3.11) has an implicit θ-dependence not controlled by the C^N closeness of the phase surfaces, Corollary 3.5 fails. Then check that the ε-removal argument, applied to the union Γ, produces (3.22) with a constant independent of θ for every q > ˜q_0. If both steps go through with explicit uniform constants, the concern is settled positively; otherwise Theorem 1.4 remains unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.4 runs through Proposition 3.1, which rests on Proposition 3.6, and in the case θ > 2N^{-1/2} the proof of (3.27) uses Corollary 3.5 to control the rescaled bilinear expression (3.32). Corollary 3.5 is exactly the uniform-in-θ bilinear estimate for the operators T_θ on the sparse time set Γ. The paper does not prove this corollary: after stating 'Corollary 3.5 can be verified by showing that Lemma 3.3 and the ε-removal argument remain valid uniformly in θ', it gives only a brief explanation that the conic surface for T_θ converges in C^N to that of T_0, citing [19, Lemma 3.1]. This is a genuine mathematical gap: the inequalities (3.10) and (3.11) in Lemma 3.3 involve a wave-packet decomposition at scale R^{-1/2} and the constants in the estimates must be independent of θ for 0 ≤ θ ≤ θ_0. Pointwise C^N convergence of the phase does not by itself guarantee that every step of the induction-on-scales proof, including the ε-removal lemma, survives with a single uniform constant. If the uniform bound in (3.22) fails, the transition from Corollary 3.5 to (3.32) is unjustified, so Proposition 3.6, Proposition 3.1, and hence Theorem 1.4 collapse. A second, smaller issue: in the proof of Proposition 3.1 the parameter ¯q_0 is chosen in (˜q_0, q_0), but for α = 1 one has ˜q_0 = q_0, so that interval is empty; this is fixable by allowing ¯q_0 up to 2(d−1+2α)/(d−1), but as written the α = 1 case is not covered by the stated argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two related problems for the circular/wave maximal operator with dilations restricted to a fractal set E ⊂ [1,2] of bounded Assouad characteristic. The first main result, Theorem 1.2, proves the missing endpoint restricted weak-type estimate for the fractal circular maximal operator in dimension d=2 at the point Q_{4,α} = (2/(2α+3), 1/(2α+3)) for 0<α≤1, complementing earlier results of Anderson–Hughes–Roos–Seeger and Roos–Seeger. The second main result, Theorem 1.4, establishes local smoothing estimates (1.4) with the sharp exponent s_c(p,q) for p,q satisfying (1.6) and q larger than an explicit threshold, extending previous ranges that were obtained via TT* arguments. The proofs combine bilinear cone restriction estimates, a sparse-time-set bilinear estimate, induction on scales, and Bourgain's summation trick. Section 4 gives independent sharpness examples showing that the exponent s_c(p,q) is necessary and that the marginal endpoint requires an epsilon loss. The paper is unconditional and contains no fitted parameters.","tokens_in":33694,"tokens_out":8876,"duration_ms":87671,"significance":"If the main theorems are correct, the paper resolves a concrete open endpoint problem posed by Roos–Seeger and extends the known range of sharp local smoothing estimates for wave propagation over fractal dilation sets. The endpoint result for the circular maximal operator is clean and likely to be influential. The sharpness constructions in Section 4 are independent and properly test the upper bounds. The main liability is that the proof of the new bilinear tool, Corollary 3.5, is only sketched and is load-bearing for Theorem 1.4; in addition, the proof of Proposition 3.1 has a small but real gap at α=1. These issues are local and appear fixable, so the work is promising but not yet in publishable form.","major_comments":[{"comment":"The uniform-in-θ bilinear estimate (3.22) is load-bearing: it is used to justify (3.32), which yields Proposition 3.6 and hence Theorem 1.4. The proof of Corollary 3.5, however, is only a sketch: it asserts that Lemma 3.3 and the ε-removal argument remain valid uniformly in θ because the phase ξ1 ϕ̃θ(ξ′/ξ1) converges to ξ1 ϕ̃0 in C^N, and it defers to [19, Lemma 3.1]. Pointwise C^N convergence of the phase does not by itself guarantee uniform constants in the wave-packet estimates (3.10)–(3.11) at scale R^{-1/2}, nor that the ε-removal lemma for the additional χ_{R^d×Γ} factor in (3.16)–(3.19) goes through with a single constant independent of θ. Since (3.22) is used directly in (3.32), the transition to Proposition 3.6 and Theorem 1.4 is unjustified without a complete proof of this uniform statement.","section":"Section 3.1, Corollary 3.5"},{"comment":"In the proof of Proposition 3.1, a pair (p*,q*) is chosen via 2/p* = q̄◦/q* and equation (3.26) for 'some q̄◦ ∈ (q̃◦, q◦)'. For α = 1 one has q̃◦ = 2(d−1+4)/(d−1+2) = 2(d+3)/(d+1) = q◦, so the interval (q̃◦, q◦) is empty and the stated argument does not cover α = 1, which is included in Theorem 1.4. This is likely fixable by treating the endpoint q̄◦ = q◦ separately or by a limiting argument, but as written the proof is incomplete for the full range 0<α≤1 claimed in the theorem.","section":"Section 3.2, proof of Proposition 3.1"},{"comment":"The ε-removal step for the sparse time set Γ is also only sketched. After asserting (3.16), the paper says it follows from 'a routine adaptation' of [9] because the χ_{R^d×Γ} factor is controlled only in terms of |E|. This is not immediate: the covering lemma from [35,36] must produce a sparse collection compatible with the interval family I in (3.3)–(3.4), and the constants must remain independent of θ when this is used for Corollary 3.5. Please provide the details of this adaptation, including how the α-sparsity of Γ enters.","section":"Section 3.1, proof of Theorem 3.2"}],"minor_comments":[{"comment":"The displayed exponent '2(1− 2α+3/q)j' is ambiguous; it should presumably be 2^{(1-(2α+3)/q)j}. Please correct the formatting and similar exponents elsewhere.","section":"Section 2, equation (2.3)"},{"comment":"The notation G is used with several meanings ('G', 'G^2', and the operator S from (2.4) and Remark 2.3). The text even notes 'we are actually abusing the notation G'. Please introduce distinct symbols or clarify the abuse precisely, especially in (3.18)–(3.19) and (3.30).","section":"Section 3.1, after (3.18)"},{"comment":"The verification that Aα(E;2^{-m}) ≲ 1 for all m ≤ j is stated as 'one can easily see'; since this uniformity is used for the sharpness results, a few more details would help the reader.","section":"Section 4, Lemma 4.1"},{"comment":"The formula for q*(α,r) is asserted after 'a computation shows'. Please include the derivation or a reference, as this formula is used to state the ε-loss extension and Remark 3.8.","section":"Section 3.3, q*(α,r)"},{"comment":"The extracted text contains several typographical artifacts (e.g., 'ESTIMA TES', 'FRACT AL', 'H¨ older', '∂α'). Please proofread the final manuscript carefully before submission.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main results are likely correct and the gaps appear fixable, but the current manuscript leaves two load-bearing steps as sketches: the uniform-in-θ bilinear estimate Corollary 3.5 and the α=1 endpoint in the proof of Proposition 3.1. Given the journal's standards, I would want to see a complete proof of Corollary 3.5 before acceptance. The paper fits the journal's scope and would be significant if the gaps are closed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it, and I largely share your assessment. The headline result is real: the d=2, α≥1/2 endpoint for the fractal circular maximal function is new, and the proof chain through bilinear estimates over sparse time sets looks structurally sound. The sharpness examples in Section 4 are independent and match the upper bounds. No sign of circularity or fitted parameters.\n\nWhat is actually new: Theorem 1.2 closes the endpoint left open in [26, Section 2.5]. The bilinear Theorem 3.2 and the induction-on-scales deduction are original, and Theorem 1.4's q-range beats [4] and [39]. The literature is cited honestly; prior results are distinguished clearly.\n\nNow the soft spots. The main one is Corollary 3.5. The paper states it 'can be verified' by adapting [40, 19, 9] uniformly in θ, then gives a brief explanation but not the details. Since Proposition 3.6 and hence Theorem 1.4 depend on the uniform bound (3.32), this is a genuine gap in the write-up. C^N convergence of the phases is plausible but doesn't automatically transfer every step of the induction-on-scales and the ε-removal lemma with constants independent of θ. I would ask a referee to see the full argument. It might be routine, but it's not in the paper. Second, smaller: in the proof of Proposition 3.1 the parameter ¯q_0 is chosen in (˜q_0, q_0), and for α=1 that interval is empty. That looks fixable (use a limit or one-sided choice), but as written it leaves α=1 unproved in Theorem 1.4's argument. Both issues are repairable, in my judgment.\n\nWho this is for: harmonic analysts working on maximal functions, local smoothing, and bilinear cone restriction. The paper is a serious piece of work and deserves a full referee if the journal can absorb a request for the missing details.\n\nI'd send it to a competent referee, not desk-reject. If I were the referee, I'd request the Corollary 3.5 proof and the α=1 fix, with a routine revision.","headline":"The endpoint theorem is real and the main proof chain appears sound; Theorem 1.4 rests on a deferred uniformity argument (Corollary 3.5) that a referee should ask to be written out, plus a small α=1 gap in the parameter choice.","tokens_in":34336,"tokens_out":5324,"would_cite":true,"duration_ms":54735,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35L05","42B20","28A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"In the plane, the fractal circular maximal function now has its missing endpoint bound at every Assouad dimension, and a sparse-time bilinear argument widens the sharp local smoothing range.","keywords":["fractal circular maximal function","spherical maximal operator","Assouad dimension","restricted weak type estimates","local smoothing","wave equation","bilinear cone restriction","fractal dilation sets"],"falsifier":"A concrete calculation: take a $2^{-j}$-separated Cantor-type set $E$ as in Lemma 4.1, with $A_\\alpha(E;2^{-j})$ and $\\tilde A_\\alpha(E;2^{-j})$ both bounded, and test the estimate (1.4) for $q$ just above the threshold in Theorem 1.4 but below the previously known range. If the estimate fails for any fixed $\\alpha\\in(0,1]$, Theorem 1.4's range is refuted; if it passes, check separately whether the uniform-in-$\\theta$ bilinear bound in Corollary 3.5 can actually be proven, since that is the step the paper leaves to a brief explanation.","tokens_in":33083,"feed_emoji":"🌀","tokens_out":8927,"duration_ms":85901,"temperature":0.7,"pith_summary":"The paper closes the last open endpoint case for the circular maximal function in two dimensions. For a dilation set $E\\subset[1,2]$ whose Assouad dimension is $\\alpha$ and whose $\\alpha$-Assouad characteristic is bounded, it proves that the maximal operator $M_E$ is of restricted weak type at the point $(1/p_\\alpha,1/q_\\alpha)=(2/(2\\alpha+3),1/(2\\alpha+3))$, the endpoint that was missing when $\\alpha\\in[1/2,1]$. It also proves a local smoothing estimate for the wave operator over fractal time sets with the sharp regularity exponent, valid for all $q>\\frac{2(d-1+2\\alpha)^2-4\\alpha^2}{(d-1)(d-1+2\\alpha)}$. If correct, the full type set of the planar circular maximal function is determined at the endpoint, and the sharp local smoothing range is strictly wider than previously known.","feed_headline":"Circular maximal function endpoint proved for all fractal dilation sets","feed_subtitle":"A bilinear sparse-time argument closes the last open case and widens sharp wave local smoothing.","key_machinery":"The load-bearing object is a bilinear restriction estimate for the cone over a sparse union of time intervals (Theorem 3.2): for caps $\\Theta,\\Theta'$ at distance $\\sim1$, the product of two adjoint restriction operators satisfies $\\|R^*f\\,R^*g\\|_{L^{q/2}(\\mathbb{R}^d\\times\\Gamma)}\\le C\\|f\\|_2\\|g\\|_2$ for $q>\\tilde q_\\circ=2(d-1+4\\alpha)/(d-1+2\\alpha)$, provided the interval collection obeys the counting bound $\\#\\{I\\in\\mathcal I:I\\cap(t,t+r)\\neq\\varnothing\\}\\le C r^\\alpha$. The proof runs an induction-on-scales argument with a wave packet decomposition at scale $R^{-1/2}$, a key lemma (Lemma 3.3) that separates the wave packets of $f$ and $g$ into aligned and non-aligned parts, and an $\\epsilon$-removal step to convert polynomial losses into the sharp exponent. A uniform-in-$\\theta$ version of the same estimate for rescaled cone operators $T_\\theta$ (Corollary 3.5) is then used, after parabolic rescaling, to turn the sparse-time bilinear bound into the linear local smoothing estimate (1.4).","core_discovery":"On the paper's own terms, the missing piece is an $L^p$-$L^q$ restricted weak-type bound for the operator $M_E$, defined by taking the supremum of circular averages over times $t\\in E$, at the boundary point $Q_{4,\\alpha}=(2/(2\\alpha+3),1/(2\\alpha+3))$ in the plane. Combining the locally constant property of the wave propagator at scale $2^{-j}$ with a bilinear restriction estimate for the cone, the proof reduces the maximal estimate to a local smoothing estimate over the $2^{-j}$-discretized set $E_j$; the bounded Assouad characteristic supplies the uniform control $A_\\alpha(E_j;2^{-j})\\le C$ that makes the $N^{1/q}$ factor harmless. Once the endpoint at $Q_{4,\\alpha}$ is available, interpolation gives $L^p\\to L^q$ boundedness on the two open segments $(Q_1,Q_{4,\\alpha})$ and $(Q_{4,\\alpha},Q_{3,\\mu})$ when the dilation set also has bounded $\\mu$-Minkowski characteristic. In the companion local smoothing result, a new bilinear restriction estimate over sparse time sets (Theorem 3.2) extends the sharp range of the estimate (1.4) with $s=s_c(p,q)$ to all $q$ above the displayed threshold, improving the previously known range from the $\\mathrm{TT}^*$ arguments of [4] and [39].","pith_inferences":["The paper's two main theorems have different security levels: Theorem 1.2 rests only on the classical bilinear cone restriction estimates [40, 34] and Bourgain's summation trick, whereas Theorem 1.4 additionally depends on the uniform-in-$\\theta$ stability of the wave packet decomposition. If forced to guess which part would survive a gap in the details, it would be Theorem 1.2.","The new bilinear estimate over sparse time sets suggests that the sharp threshold for Conjecture 1.3, namely $q=2(d-1+2\\alpha)/(d-1)$, is a Knapp-type obstruction rather than a technical one; a natural test is whether multilinear refinements can reach it.","The paper's use of $\\tilde A_\\alpha(E;\\delta)$, a stronger quantitative control than the bounded Assouad characteristic, indicates that endpoint local smoothing may require uniform control across all scales; Theorem 1.4's range may not be optimal for sets with only bounded Assouad characteristic.","Because the proof of Theorem 1.2 does not use the stronger $\\tilde A_\\alpha$ condition, the endpoint result for the circular maximal function is more robust than the local smoothing theorem, which may still admit an extension by removing the uniform-in-scale assumption."],"forward_implications":["The missing endpoint at $Q_{4,\\alpha}$ now holds for every $0<\\alpha\\le1$ in the plane, so the type set of the circular maximal function is known on the boundary segment through $Q_{4,\\alpha}$ whenever $E$ has bounded $\\alpha$-Assouad characteristic.","Adding a bounded $\\mu$-Minkowski characteristic, the open line segments $(Q_1,Q_{4,\\alpha})$ and $(Q_{4,\\alpha},Q_{3,\\mu})$ lie inside the $L^p\\to L^q$ boundedness range.","The local smoothing estimate (1.4) holds with the optimal regularity exponent $s_c(p,q)$ for all $q$ above the threshold in Theorem 1.4, improving on the ranges obtained in [4] and [39].","With an $\\epsilon$-loss in regularity, the range extends further, and in dimensions $d\\ge4$ the loss can be removed using the known optimal local smoothing estimate for $q$ above $r_d=2+4/(d-3)$.","If Conjecture 1.3 is correct, the remaining gap is only the limiting case $q=2(d-1+2\\alpha)/(d-1)$, where the paper itself shows the estimate fails without an additional $2^{\\epsilon j}$ loss."],"supporting_citations":[{"why":"Supplies Theorem 1.1, the earlier endpoint estimates that leave open only the case $Q_{4,\\alpha}$ for $d=2,\\alpha\\ge1/2$.","marker":"[1]"},{"why":"Establishes the sharp range for the fractal circular maximal function and explicitly raises the endpoint problem solved here.","marker":"[26]"},{"why":"Provides the sharp bilinear cone restriction estimate that is the engine of Proposition 2.1 (Theorem 2.5).","marker":"[40]"},{"why":"Adds the endpoint case $q=q_\\circ$ of the bilinear cone restriction estimate used in Proposition 2.1.","marker":"[34]"},{"why":"Contributes the circular maximal endpoint results and Bourgain's summation trick (Lemma 2.2) on which the proof of Theorem 1.2 relies.","marker":"[18]"},{"why":"Supplies the induction-on-scales framework and wave packet decomposition that Theorem 3.2 adapts to sparse time sets.","marker":"[19]"},{"why":"Gives the $\\epsilon$-removal lemma that converts the induction-on-scales bounds with polynomial losses into the claimed sharp exponents.","marker":"[9]"},{"why":"Earlier local smoothing result over fractal time sets, obtained by a $\\mathrm{TT}^*$ argument; Theorem 1.4 extends the range of $q$ for which the optimal exponent holds.","marker":"[4]"},{"why":"Independent earlier local smoothing result over fractal time sets whose range Theorem 1.4 also extends.","marker":"[39]"},{"why":"Optimal high-dimensional local smoothing estimate used in Remark 3.8 to remove the $\\epsilon$-loss in dimensions $d\\ge4$.","marker":"[16]"}],"fun_headline_variants":["Endpoint for fractal circular maximal function finally proved","Bilinear route to fractal circular maximal endpoint","Sharp endpoint for fractal circular maximal function","Fractal circular maximal endpoint settled","Circular maximal endpoint closed for all fractal dimensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Theorem 1.4 rests on Corollary 3.5, which asserts that the key decomposition lemma and the $\\epsilon$-removal argument from [9, 19, 40] remain valid uniformly in the rescaling parameter $\\theta$ for the operators $T_\\theta$; the paper states this can be verified but gives only a brief explanation, so a failure of that uniform stability would break the transfer of the sparse-time bilinear estimate to the rescaled operators and with it Proposition 3.1 and Theorem 1.4.","fun_headline_variants_meta":{"raw":{"variants":["Endpoint for fractal circular maximal function finally proved","Bilinear route to fractal circular maximal endpoint","Sharp endpoint for fractal circular maximal function","Fractal circular maximal endpoint settled","Circular maximal endpoint closed for all fractal dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000583,"raw_usage":{"total_tokens":2782,"prompt_tokens":1022,"completion_tokens":1760,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":1695}},"tokens_in":638,"tokens_out":1760,"duration_ms":14949,"temperature":1.0,"reasoning_tokens":1695,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:49:20.186536+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete calculation: take a $2^{-j}$-separated Cantor-type set $E$ as in Lemma 4.1, with $A_\\alpha(E;2^{-j})$ and $\\tilde A_\\alpha(E;2^{-j})$ both bounded, and test the estimate (1.4) for $q$ just above the threshold in Theorem 1.4 but below the previously known range. If the estimate fails for any fixed $\\alpha\\in(0,1]$, Theorem 1.4's range is refuted; if it passes, check separately whether the uniform-in-$\\theta$ bilinear bound in Corollary 3.5 can actually be proven, since that is the step the paper leaves to a brief explanation.","supporting_citations":[{"cited_title":"Anderson, K","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 1.1, the earlier endpoint estimates that leave open only the case $Q_{4,\\alpha}$ for $d=2,\\alpha\\ge1/2$."},{"cited_title":"Roos and A","cited_arxiv_id":null,"evidence_quote":"Establishes the sharp range for the fractal circular maximal function and explicitly raises the endpoint problem solved here."},{"cited_title":"Wolff, A sharp bilinear cone restriction estimate , Ann","cited_arxiv_id":null,"evidence_quote":"Provides the sharp bilinear cone restriction estimate that is the engine of Proposition 2.1 (Theorem 2.5)."},{"cited_title":"Tao, Endpoint bilinear restriction theorems for the cone, and some sharp null form esti- mates, Math","cited_arxiv_id":null,"evidence_quote":"Adds the endpoint case $q=q_\\circ$ of the bilinear cone restriction estimate used in Proposition 2.1."},{"cited_title":"Lee, Endpoint estimates for the circular maximal function, Proc","cited_arxiv_id":null,"evidence_quote":"Contributes the circular maximal endpoint results and Bourgain's summation trick (Lemma 2.2) on which the proof of Theorem 1.2 relies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the induction-on-scales framework and wave packet decomposition that Theorem 3.2 adapts to sparse time sets."},{"cited_title":"Bourgain and L","cited_arxiv_id":null,"evidence_quote":"Gives the $\\epsilon$-removal lemma that converts the induction-on-scales bounds with polynomial losses into the claimed sharp exponents."},{"cited_title":"Variation bounds for spherical averages over restricted dilates","cited_arxiv_id":"2409.05579","evidence_quote":"Independent earlier local smoothing result over fractal time sets whose range Theorem 1.4 also extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Optimal high-dimensional local smoothing estimate used in Remark 3.8 to remove the $\\epsilon$-loss in dimensions $d\\ge4$."}],"review_version":1}