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Fourier Frames on Salem Measures

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.01280 v1 pith:WFEA2QLU submitted 2025-06-02 math.CA math.FA

Longhui Li , Bochen Liu This is my paper · ORCID
classification math.CAmath.FA MSC 28A7842C1528A8011J83
keywords FourierframesSalemmeasuresdimensionframe-spectralCantorBrownianimagesDiophantineapproximationexponentialbases
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [§2.2, Proposition 2.2; also §4 and §5] 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.
  2. [§9, equation (9.12)] 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).
  3. [§5, Theorem 5.1] 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].
minor comments (5)
  1. [§4, Theorem 4.1] The notation 'µ(x0, r)' should be 'µ(B(x0, r))'; the same correction is needed in the verification paragraph after the construction.
  2. [§3, last paragraph of Section 3] 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.
  3. [§2.2, Proposition 2.1 proof] There is a typo in the proof: 'j′,, . . . , j′_N' should be 'j'_1, . . . , j'_N'.
  4. [§10] The parenthetical 'C^∞_0' should be 'C^∞_c' or 'C^∞_0' consistently with the rest of the paper.
  5. [§5, (5.13)] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity: central claim is overdetermined and self-contained; the only author-overlap citations are non-load-bearing.

full rationale

No circular reduction is exhibited. Theorem 1.1 is overdetermined: the one-line construction (2.2) already yields an s-dimensional Salem measure without Fourier frames from any s-dimensional Salem measure ν via the external Dutkay–Lai uniformity criterion, and Sections 3–9 give independent generic constructions (convolution, non-convolution, Brownian image, deterministic Diophantine). The author-overlap citations are not load-bearing. [21] is a published criterion whose proof is adapted in Proposition 2.2 under explicit assumptions (Fourier decay of a nonzero f dµ and Frostman lower bound); it is not a restatement of the target result. [34] is explicitly declared insufficient ('this is not enough for our use') and the analogous Frostman estimate is reproved in Lemma 7.1 rather than imported. Lemma 8.2 is derived in-paper from the explicit nested prime sets P_i^ν ⊂ P_i^µ, the pointwise coefficient comparison (8.1), and the disjoint-support estimate (7.6); the Section 9 contradiction compares an upper counting bound with independent lower bounds on bµ(k), not with the desired conclusion. The cF^ν_i symbols in the display just before (9.12) are an apparent typo (the adjacent text invokes cF^µ_i from (6.4) and positivity of cF^µ_i); this is a typographical slip, not a circular step. Overall, the derivations are self-contained given standard theorems, so the appropriate score is 2 at most, reflecting only minor non-load-bearing self-citations.

Assumptions & free parameters 6 free parameters · 6 assumptions · 2 invented entities

All parameters are construction constants chosen to make inequalities work, not fitted to data. The paper imports standard analytic and probabilistic theorems (Prime Number Theorem, Bernstein's inequality, Kahane's Brownian image theorem, Dutkay-Lai's criterion, Shi's counting lemma) as external benchmarks; the new content is the combination of these with new criteria (Proposition 2.2) and auxiliary objects.

free parameters (6)
  • d_k, r_k, L_k in Section 3 = d_k=k+1, r_k=sqrt(log k), L_k ≈ d_k^{-1/s}
    Chosen so that the convolution construction satisfies Lemma 3.2's separation condition and carries Salem dimension s.
  • Weights p_{j,k} in Section 3 = p_{1,k}=(1-1/(k+1))/d_k, p_{2,k}=(1+1/(k+1))/d_k, p_{j,k}=1/d_k for j≥2
    Chosen so the max/min product of joint weights diverges, satisfying Proposition 2.1 while preserving the Fourier-decay argument.
  • Unequal-splitting constant in Sections 4-5 = 1/sqrt(2)
    Chosen as the largest value that keeps the modified Chen-type measure s-dimensional Salem while creating heavy intervals of mass r^{s/2}.
  • Rapid growth rate q_i = q_{i+1} ≥ q_i^{10}
    Makes error terms and infinite products in Sections 6-9 converge or diverge as needed.
  • Auxiliary integer function h(i) = h(i)=C_s log q_i
    Chosen to balance convergence of ∏(#P_i^µ/#P_i^ν) in Lemma 8.1 with divergence of n log q_n ∏_{i<n} q_i log q_i / ∏ h(i) in the final contradiction.
  • Auxiliary bump function ϕ = Explicit combination in Section 6.2 with bϕ ≈ (1+|ξ|)^{-4}
    Constructed by hand to be nonnegative, supported in (-1,1), with nonnegative and slowly varying Fourier transform; this guarantees bµ(k)>0.
assumptions (6)
  • standard math Prime Number Theorem for counting primes up to q^{s/2} in dyadic ranges
    Used throughout Sections 6-9 to count #{p≤X: prime} and to estimate Fourier coefficients of F_i^µ and F_i^ν (e.g., Eq (6.10), (7.4)).
  • standard math Bernstein's inequality for sums of independent bounded random variables (Lemma 4.2)
    Used in Section 4 to prove the existence of good random choices A_j yielding the Fourier-decay bound (4.6).
  • standard math Kahane's theorem on Fourier decay of Brownian images of Frostman measures
    The starting point of Section 5; the paper extends it by replacing the classical ball condition (5.1) with the weaker conditions (5.3)-(5.4).
  • standard math Dutkay-Lai uniformity criterion for non-existence of Fourier frames
    Used for the one-line construction (2.2) and as the basis of Proposition 2.1.
  • standard math Shi's counting lemma relating frame-spectrum density to local mass
    Cited and used inside the proof of Proposition 2.2 to convert a Fourier-decay divergence into a counting contradiction.
  • standard math Energy/Hausdorff dimension equivalence from Frostman's lemma (Eq (2.1))
    Used to read off Hausdorff dimension from Fourier decay when proving that each construction is Salem.
invented entities (2)
  • Auxiliary measure ν on Diophantine approximation sets
    purpose: Supports the proof that the target measure µ admits no Fourier frame: ν is used in Lemma 8.1-8.2 and Section 9 to derive a counting contradiction from the frame inequalities.
    ν is constructed only as a proof device (Section 7); it makes no predictions outside the paper and is not claimed to be observable.
  • Auxiliary bump function ϕ with nonnegative Fourier transform
    purpose: Ensures µ has positive integer Fourier coefficients and enables the comparison |bν(k+l)| ≤ C bµ(k)+Cε(1+|k|)^{-1+ε} in Lemma 8.2.
    An explicit admissible ϕ is built in Section 6.2; it is a technical lever, not an empirical entity.

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Cite this review

Pith. "Pith review of Fourier Frames on Salem Measures." pith.science (2026). https://pith.science/paper/WFEA2QLU

@misc{pith2026250601280,
  author       = {Pith},
  title        = {Pith review of: Fourier Frames on Salem Measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WFEA2QLU}},
  note         = {Machine review of arXiv:2506.01280}
}
abstract

For every $0<s\leq 1$ we construct $s$-dimensional Salem measures in the unit interval that do not admit any Fourier frame. Our examples are generic for each $s$, including all existing types of Salem measures in the literature: random Cantor sets (convolutions, non-convolutions), random images, and deterministic constructions on Diophantine approximations. They even appear almost surely as Brownian images. We also develop different approaches to prove the nonexistence of Fourier frames on different constructions. Both the criteria and ideas behind the constructions are expected to work in higher dimensions. On the other hand, we observe that a weighted arc in the plane can be a $1$-dimensional Salem measure with orthonormal basis of exponentials. This leaves whether there exist Salem measures in the real line with Fourier frames or even orthonormal basis of exponentials a subtle problem.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Walsh-Quotient Obstruction for Fourier Frames on Odd Reciprocal-Power Bernoulli Convolutions

    math.FA 2026-07 unverdicted novelty 7.5 of 10

    Odd reciprocal-power Bernoulli convolutions admit no Fourier frames in L² when the contraction ratio is the reciprocal of an odd integer power.

  2. Cantor measures with odd base do not admit Fourier frames

    math.FA 2026-07 accept novelty 7.0 of 10 full

    Cantor measures with odd integer base b > 1 do not admit Fourier frames, answering Strichartz's question for the middle-third Cantor measure.

  3. Fourier frames on smooth surfaces with nonvanishing Gaussian curvature

    math.CA 2025-07 accept novelty 7.0 of 10

    Compact smooth curved surfaces with nonzero Gaussian curvature, including hemispheres and self-intersecting curves, admit no Fourier frames.

Reference graph

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