{"id":"ff0f325b-db6e-43ac-8afe-fb4cd67406f3","arxiv_id":"2506.01280","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"For every 0<s≤1 there are s-dimensional Salem measures on the unit interval admitting no Fourier frame, and such measures appear in every known Salem construction type.","lead":"This paper constructs, for every dimension s between 0 and 1, s-dimensional Salem measures on the unit interval that admit no Fourier frame. The examples come from all known Salem construction methods, including random Cantor sets, Brownian images, and Diophantine approximation sets.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; Theorem 1.1 is over-determined by independent constructions, and the apparent cFν/cFμ typo in Eq. (9.12) is cosmetic.","rationale":"The reader's verdict of ACCEPT with moderate confidence is appropriate. I focused on the Section 9 Diophantine construction because it is the most intricate and because Lemma 8.2 is genuinely delicate. The pointwise comparison (8.1) is sound when C_s in h(i)=C_s log q_i is chosen small enough: the correction term A satisfies A≤#Pν_i/(q_i^{s/2}/h(i)-1)≈C_s, so with C_s<1/c_s one gets A≤1, and the nested prime sets then give |Nν-A|≤Nν+1≤#{p∈Pμ_i:p|m}. The decomposition into I and II in Lemma 8.2 is a correct partition of the convolution coefficient (the m_{n0}=-l terms are exactly the II sum), and the decay estimates are consistent with the rapid growth q_{n+1}≥q_n^{10n}. In Section 9, the tail of the frame sum over |λ|>4q_n tends to zero by (9.1) and (9.3), and the Frostman bound from Lemma 7.1 combined with the lower bound on bμ(k) produces the advertised contradiction. The only flaw I noticed is the cFν_i/cFμ_i typo in (9.12); the very next sentence in the paper refers to the expression for cFμ_i, so the intended inequality is clear and repairable. Since Theorem 1.1 is also supported by the one-line construction, Theorem 3.1, Theorem 4.1, and Theorem 5.1, no single possible defect in the Diophantine section is load-bearing for the paper's main claim. The paper is not machine-checked, and a fully independent check of Lemma 8.2 would still be worthwhile, but at this review level the central argument holds.","tokens_in":29946,"tokens_out":35365,"duration_ms":359123,"concrete_test":"Verify Eq. (9.12) with the product corrected from cFν_i to cFμ_i. Check that for each |k|≤4q_n and every tuple k_1+...+k_n=k with |k_i|≤q_i for i<n, one has |k_n|≤5q_n, so cFμ_n(k_n)≥c q_n^{-s/2}\\log q_n uniformly; then the remaining sum over k_1,...,k_{n-1} factorizes as ∏_{i=1}^{n-1}∑_{|k_i|≤q_i}cFμ_i(k_i), giving the stated lower bound. If this factorization holds, the Section 9 contradiction is sound; if not, the repair is localized to the Diophantine-genericity claim and does not affect Theorem 1.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central existence claim, Theorem 1.1, does not rest on a single fragile argument: s-dimensional Salem measures without Fourier frames are obtained independently by the one-line construction (2.2) via the Dutkay–Lai uniformity criterion, by the convolution construction in Theorem 3.1, by the non-convolution construction in Theorem 4.1, and by the Brownian-image theorem 5.1. Even if one of these routes were defective, the others would still deliver the conclusion for every 0<s≤1. The most delicate part is the deterministic Diophantine argument in Sections 6–9, and Lemma 8.2 is intricate, but I found no internal inconsistency that breaks the argument. The lower-bound display (9.12) contains an apparent typo, using cFν_i(k_i) where the surrounding text and subsequent estimates use cFμ_i(k_i); however, the intended lower bound is valid with cFμ_i, since cFμ_i(k_i)≥0, cFμ_i(0)=1, and for nonzero |k_i|≲q_i one has cFμ_i(k_i)≳q_i^{-s/2}\\log q_i. This is a typographical slip, not a load-bearing mathematical gap. The rest of the Section 9 counting argument, including the use of (8.1), the Frostman estimate in Lemma 7.1, and the separation into I and II in Lemma 8.2, checks out under the stated choice h(i)=C_s\\log q_i with C_s sufficiently small.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for every 0 < s ≤ 1 there exist s-dimensional Salem measures on [0,1] that admit no Fourier frame. The result is obtained through several independent routes: a one-line construction from an arbitrary Salem measure using the Dutkay-Lai uniformity criterion; a modified Salem infinite-convolution construction; a non-convolution random Cantor construction; almost sure Brownian images; and a deterministic Kaufman-type Diophantine construction that occupies about half the paper. The authors also show that a weighted arc in the plane is a 1-dimensional Salem measure with an orthonormal basis of exponentials, thereby sharpening the contrast between the real-line and higher-dimensional settings.","tokens_in":30372,"tokens_out":48523,"duration_ms":453248,"significance":"If the main theorem is correct, it settles a basic structural question: maximal Fourier decay does not force frame-spectrality for measures on the real line. The paper is particularly valuable because the nonexistence is demonstrated for essentially every known type of Salem measure construction, including almost-sure Brownian images, and because the Diophantine argument develops a new technique (comparison with an auxiliary measure with positive Fourier coefficients) that is likely to generalize to higher dimensions. The convolution construction is also essentially Ahlfors-David regular, which shows the phenomenon is not caused by measure irregularity. The paper is clearly written and the main inequalities are plausible and largely verifiable in detail. However, several statements in the central criteria and in the final Diophantine lower bound contain quantifier or notational errors that must be corrected before the paper is publication-ready.","major_comments":[{"comment":"Condition (ii) in Proposition 2.2, stated as sup_x μ(B(x,r)) ≳ r^α for all r > 0, cannot hold for any compactly supported finite measure with α > 0: for r at least the diameter of the support, the left-hand side equals the total mass, which is constant, while the right-hand side grows without bound. The proof only needs the lower bound for small r, since it is applied with r = (10R)^{-1} for arbitrarily large R. Please change the quantifier to 'for all sufficiently small r' (or 'for all 0 < r < r_0') in Proposition 2.2, in the statements of Theorems 4.1 and 5.1, and in the verification paragraphs that follow. As written, the proposition is vacuous for compactly supported probability measures, and the applications in Sections 4 and 5 are not formally justified.","section":"§2.2, Proposition 2.2; also §4 and §5"},{"comment":"The displayed lower bound for bµ(k) starts with products cF_i^ν(k_i), but the convolution expansion of bµ(k) is in terms of cF_i^μ, and cF_i^ν can be negative, so the inequality as written is not justified. The next sentence explicitly invokes the expression of cF_i^μ, indicating that the ν superscripts in (9.12) are typographical errors. Please replace every cF_i^ν in this display by cF_i^μ; with that correction the lower bound is valid, because the cF_i^μ are nonnegative and the restricted partition sums are part of the expansion of bµ(k).","section":"§9, equation (9.12)"},{"comment":"The conclusion 'µω([0,r]) ≥ µ([0,Cω r^{1/α}))' is not the correct statement for Brownian images. A Brownian path can take negative values, so the preimage of [0,r] need not contain [0, C r^{1/α}]. The argument actually gives containment ω([0,t]) ⊂ [-C t^α, C t^α], hence a lower bound for µω(B(0,r)) (equivalently µω([-r,r])). Please restate the theorem and the application to Proposition 2.2 using centered balls rather than the interval [0,r].","section":"§5, Theorem 5.1"}],"minor_comments":[{"comment":"The notation 'µ(x0, r)' should be 'µ(B(x0, r))'; the same correction is needed in the verification paragraph after the construction.","section":"§4, Theorem 4.1"},{"comment":"The phrase 'dµt = ψ dµt' appears to be a typo and should read something like 'dµt = ψ dµt' with ψ equal to 1 on the support, or more clearly 'dµt = ψ dν' for a suitable smooth ψ; please clarify.","section":"§3, last paragraph of Section 3"},{"comment":"There is a typo in the proof: 'j′,, . . . , j′_N' should be 'j'_1, . . . , j'_N'.","section":"§2.2, Proposition 2.1 proof"},{"comment":"The parenthetical 'C^∞_0' should be 'C^∞_c' or 'C^∞_0' consistently with the rest of the paper.","section":"§10"},{"comment":"The change of variables leading from the double integral to the integral over r is correct but quite compressed; a one-line derivation of the identity ∫_0^1 1_{t ≤ a^{-1}log r^{-1}} dr = e^{-at} would help the reader.","section":"§5, (5.13)"}],"recommendation":"major_revision","confidential_remarks":"The central claim is likely correct and is over-determined by the independent constructions, so rejection is not warranted. The main work for revision is local: repair the impossible universal quantifier in Proposition 2.2 and the analogous statements, correct the ν/μ superscripts in (9.12), and restate the Brownian theorem with centered balls. The self-citations to [21] and [34] do not appear circular: they are used for auxiliary criteria or context, not for the main conclusion. I would be comfortable accepting after these corrections are made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: Theorem 1.1 is the genuinely new result, and it is supported by four independent construction routes, so even if one of them cracks, the conclusion survives. The paper does more than fill the missing case: it proves a clean anti-frame criterion (Proposition 2.2), shows how to adapt Salem's original convolution construction to force frame failure, and establishes that Brownian images of certain measures are almost surely Salem without frames. The one-line construction in (2.2) is elementary, but the authors are upfront that it is just a warm-up; the real work is showing that the phenomenon is not an artifact of measure surgery but appears in every known family of Salem measures.\n\nThe soft spots are where you would expect them. Sections 6-9, the deterministic Diophantine construction, are intricate and deserve a second pair of eyes. Lemma 8.2 rests on a sequence of structural coincidences: the auxiliary function with nonnegative, slowly varying Fourier transform, the nested prime sets, and the disjoint-support estimate (7.6). If any of those failed, the counting argument in Section 9 would collapse. I did not find an actual contradiction, and the stress-test note is right that the apparent cFν/cFμ slip in (9.12) is a typo, not a gap; substituting cFμ gives the claimed lower bound. The modularity is a real virtue: even if the Diophantine route needed repair, the convolution and Brownian-image routes already cover every 0<s≤1.\n\nOne caveat on the word 'generic': the paper shows the phenomenon across all existing Salem constructions, not a single Baire-category or measure-one statement in the space of all s-dimensional Salem measures. The Brownian-image and random Cantor statements are almost-sure, so they are strong genericity assertions. I do not think the wording is misleading, but a reader should not expect one all-encompassing generic statement.\n\nThe citation pattern is fine. [21] and [34] overlap with the authors, but neither supplies the central conclusion; the main derivations are self-contained. The planar arc example is a nice counterpoint and shows why the real-line problem is truly subtle.\n\nThis paper deserves a serious referee. I would send it out.","headline":"A genuine resolution of the missing real-line case, with four independent construction routes; the Diophantine part is dense but I see no fatal gap.","tokens_in":30862,"tokens_out":2066,"would_cite":true,"duration_ms":21807,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28A78","42C15","28A80","11J83"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every 0 < s ≤ 1 there exist s-dimensional Salem measures on the unit interval that admit no Fourier frame, and the phenomenon is generic across all known Salem constructions.","keywords":["Fourier frames","Salem measures","Fourier dimension","frame-spectral measures","Cantor measures","Brownian images","Diophantine approximation","exponential bases"],"falsifier":"Compute, for the explicit φ, the rapid sequence q_i, and the nested prime sets in Sections 6–8, the quantities |bν(k + l)| and bµ(k) at integers k with |k| just above 2q_n and |l| near q_n/2: if the inequality |bν(k + l)| ⩽ C bµ(k) + Cε(1 + |k|)^{−1+ε} ever fails with the stated constants, the main contradiction collapses. Alternatively, for any constructed measure, exhibit a countable set Λ and constants 0 < A ⩽ B < ∞ satisfying the frame inequality for all f ∈ L²(µ); the theorem predicts no such Λ exists.","tokens_in":29775,"feed_emoji":"📐","tokens_out":10901,"duration_ms":94549,"temperature":0.7,"pith_summary":"This paper proves that a Salem measure—a measure whose Fourier transform decays as fast as its dimension allows—need not admit any Fourier frame on the real line. For every 0 < s ⩽ 1 it constructs s-dimensional Salem measures supported on the unit interval for which no countable set of exponentials satisfies the frame inequality. These examples are not isolated: they arise generically from every existing type of Salem construction, including random Cantor sets (both convolution and non-convolution types), random images, deterministic constructions on Diophantine approximations, and almost surely as images of Brownian motion. The paper also observes that a weighted arc on the unit circle is a 1-dimensional Salem measure with an orthonormal basis of exponentials, which leaves the existence of any Salem measure on the real line with a Fourier frame a subtle open problem.","feed_headline":"Salem measures can refuse every Fourier frame","feed_subtitle":"For every dimension up to 1, the authors build them from fractals, random images, and Brownian motion.","key_machinery":"Three distinct mechanisms do the work. The convolution case (Proposition 2.1) uses a uniformity criterion: if an infinite convolution has weights whose product ratios diverge, any frame spectrum would force a contradiction from the ratio of localized frame sums on translated copies of the measure. The random non-convolution and image cases use a new criterion (Proposition 2.2): a measure with |bµ(ξ)| ≲ |ξ|^{−β/2} and a ball of mass at least r^α at every scale, with α < β, admits no Fourier frame, because the frame inequality plus the heavy ball forces the spectrum to grow at most like r^α while the Fourier decay forces ∑_{λ≠0}|λ|^{−β} = ∞. The deterministic Diophantine case is the deepest: it builds an auxiliary measure ν ≪ µ with bounded density and the pointwise Fourier-coefficient estimate |bν(k + l)| ⩽ C bµ(k) + Cε(1 + |k|)^{−1+ε} uniformly for |l| ≲ |k| (Lemma 8.2), supported by a specially chosen φ with nonnegative slowly varying Fourier transform, nested prime sets P_i^ν ⊂ P_i^µ, and a disjoint-support estimate from excluding pZ. This comparison lets the authors transplant a frame from µ to ψν, average over intervals, and obtain a counting lower bound from the Frostman property of ν that contradicts the upper bound from the frame inequality.","core_discovery":"The central claim is Theorem 1.1: for every 0 < s ⩽ 1 there exist s-dimensional Salem measures on the unit interval that do not admit any Fourier frame, and such measures are generic for each s in the sense that they emerge from all known Salem constructions. Section 3 modifies the original convolution construction to make the weight ratios diverge, so the uniformity criterion rules out frames while the Fourier decay is preserved. Section 4 modifies a non-convolution random Cantor construction to produce a measure with Fourier decay exponent s/2 while having an interval of mass about $r^{{s/2}}$ at every scale, so the new Proposition 2.2 rules out frames. Section 5 proves that Brownian images of suitable input measures are Salem measures without frames almost surely. Sections 6–9 handle the deterministic Diophantine-approximation measures, constructing an auxiliary measure ν absolutely continuous with respect to the target µ and with Fourier coefficients pointwise controlled by µ's, and deriving a counting contradiction that rules out any frame spectrum. Finally, Section 10 shows a weighted arc in the plane is a 1-dimensional Salem measure with an orthonormal basis of exponentials.","pith_inferences":["The paper leaves implicit that the pointwise Fourier-coefficient comparison of Lemma 8.2 may serve as a template for other Kaufman-type deterministic measures with positive Fourier coefficients; a natural test is whether the very recent higher-dimensional constructions mentioned in Section 2.3 inherit the same frame-free property.","A natural extension would be to make the 'generic for each s' statement precise as Baire-generic or in the sense of random constructions, and to check whether typical Salem measures on [0,1] admit no Fourier frame at all; the paper's examples strongly suggest but do not prove such a statement.","The input measure µ in the Brownian-image theorem is required to have one-sided ball estimates (5.3)–(5.5); a testable variation is to replace these by two-sided estimates and see whether the Fourier decay exponent or the frame obstruction changes.","If the open problem is resolved positively, the planar-arc example suggests the spectral Salem measure would need some curvature-like structure absent in the line; if resolved negatively, the present examples would be the first of a general real-line phenomenon."],"forward_implications":["Frame-spectrality is not a consequence of maximal Fourier decay on the real line: a Salem measure can have Fourier dimension equal to its Hausdorff dimension and still admit no Fourier frame.","The failure is generic across all known Salem constructions, and the Brownian-image examples show such measures occur almost surely, not just by sparse or artificial choices.","The new criteria are designed to lift to higher dimensions, so Salem measures without Fourier frames should exist in R^d for every d ≥ 1.","Because every subset of finite Lebesgue measure admits Fourier frames, these examples isolate singular measures as the source of the obstruction.","The planar weighted arc shows a Salem measure can be spectral in the plane, so the real-line question—whether any Salem measure on R has a Fourier frame—remains genuinely open."],"supporting_citations":[{"why":"Supplies the uniformity criterion for infinite convolutions used in Proposition 2.1.","marker":"[9]"},{"why":"Supplies the surface-measure nonexistence argument and the localized average estimates adapted here.","marker":"[21]"},{"why":"Supplies the Fourier-decay estimate for images of Brownian motion that Theorem 5.1 extends.","marker":"[24]"},{"why":"Supplies the Diophantine-approximation construction that Sections 6–9 build on.","marker":"[27]"},{"why":"Supplies the Frostman estimate and dimension bounds for Diophantine-approximation measures used for the auxiliary measure ν.","marker":"[34]"},{"why":"Supplies the classical proof that ball conditions give Fourier decay under Brownian images, which Theorem 5.1 relaxes.","marker":"[35]"},{"why":"Supplies the infinite-convolution construction and its key Fourier-decay lemma, modified in Section 3.","marker":"[37]"},{"why":"Supplies the non-convolution random Cantor construction that Section 4 modifies.","marker":"[5]"},{"why":"Supplies a Fourier-decay-to-nonexistence criterion that motivates the new Proposition 2.2.","marker":"[33]"},{"why":"Supplies the ball-counting estimate used inside Proposition 2.2.","marker":"[38]"}],"fun_headline_variants":["Generic Salem measures admit no Fourier frame","For every s≤1, a Salem measure with no Fourier frame","Brownian images and Cantor sets yield frame-free Salem measures","No Fourier frame for generic Salem measures in any dimension","Fourier frame nonexistence for Salem measures at every s≤1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The final contradiction assumes a specially built helper measure whose Fourier coefficients stay pointwise below the original measure's at every relevant frequency; if that comparison fails at any scale, the counting argument that rules out frames collapses.","fun_headline_variants_meta":{"raw":{"variants":["Generic Salem measures admit no Fourier frame","For every s≤1, a Salem measure with no Fourier frame","Brownian images and Cantor sets yield frame-free Salem measures","No Fourier frame for generic Salem measures in any dimension","Fourier frame nonexistence for Salem measures at every s≤1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000997,"raw_usage":{"total_tokens":4212,"prompt_tokens":924,"completion_tokens":3288,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":3206}},"tokens_in":540,"tokens_out":3288,"duration_ms":24197,"temperature":1.0,"reasoning_tokens":3206,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:47:54.149055+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for the explicit φ, the rapid sequence q_i, and the nested prime sets in Sections 6–8, the quantities |bν(k + l)| and bµ(k) at integers k with |k| just above 2q_n and |l| near q_n/2: if the inequality |bν(k + l)| ⩽ C bµ(k) + Cε(1 + |k|)^{−1+ε} ever fails with the stated constants, the main contradiction collapses. Alternatively, for any constructed measure, exhibit a countable set Λ and constants 0 < A ⩽ B < ∞ satisfying the frame inequality for all f ∈ L²(µ); the theorem predicts no such Λ exists.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Fourier-decay estimate for images of Brownian motion that Theorem 5.1 extends."},{"cited_title":"Iosevich, C.-K","cited_arxiv_id":null,"evidence_quote":"Supplies the surface-measure nonexistence argument and the localized average estimates adapted here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Diophantine-approximation construction that Sections 6–9 build on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical proof that ball conditions give Fourier decay under Brownian images, which Theorem 5.1 relaxes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the infinite-convolution construction and its key Fourier-decay lemma, modified in Section 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the non-convolution random Cantor construction that Section 4 modifies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies a Fourier-decay-to-nonexistence criterion that motivates the new Proposition 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the ball-counting estimate used inside Proposition 2.2."}],"review_version":1}