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A measure with small support and p-summable Fourier transform

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Pith's one-line read The paper constructs a probability measure whose Fourier transform lies in $L^p(\mathbb{R}^d)$ for any $2<p<\infty$, even though its support is a compact set of zero $2d/p$-dimensional Hausdorff measure; this refutes the endpoint form of…

desk verdict Settles the endpoint Hausdorff-measure uncertainty principle with a branch-dependent random Cantor construction; the proof is sound and the only issues are presentation-level gaps. read the letter →

arxiv 2412.07314 v1 pith:VWXOXEUP submitted 2024-12-10 math.CA

Nikita P. Dobronravov This is my paper · ORCID
classification math.CA MSC 42B1028A7828A80
keywords uncertaintyprincipleFouriertransformHausdorffmeasureCantor-typesetp-summableL^pintegrabilityrandomconstructionpacking
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 the endpoint form of the classical uncertainty principle fails for general compact sets. For any $2

What carries the argument

The central object is the Cantor-type set $C=\cap_n C_n$ built from an infinite tree $T$: vertices are cubes, each cube $Q_k$ is assigned weight $b(Q_k)=\prod M_j^{-1}$ over its ancestors, and a child $Q_j$ of $Q_k$ has side length $l(Q_j)=M_k^{-p/(2d)}b(Q_k)^{p/(2d)}/(n(Q_k)+1)$. This side-length formula (2.5) is the load-bearing identity: it makes the Hausdorff covering sum over the $n$-th layer equal to $1/n^{2d/p}$, forcing $H^{2d/p}(C)=0$, and it makes the $L^p$ increment of the Fourier transform of the $k$-th approximating measure comparable to $b(Q_k)^{p/2}(n(Q_k)+1)^d$, whose sum over all layers converges because $b(Q_k)\le 2^{-n}$ and $p>2$. The second ingredient is the randomized measure $\nu_{M,r}=M^{-1}\sum_{j=1}^M S_j \lambda_{[0,r]^d}$, whose expectation and variance in Fourier space are controlled by Lemmas 3.1 and 3.2; a realization selected in Corollary 3.3 keeps the deviations small enough for Lemmas 3.6 and 3.7 to close the argument.

What would settle it

Compute the covering sum in Proposition 2.1 for the explicit side lengths (2.5) and verify that the sum of $\operatorname{diam}(Q)^{2d/p}$ over all cubes in the $n$-th layer equals $1/n^{2d/p}$ up to a universal constant, and check that the layer sum in (3.18) converges exactly for $p>2$; if either fails, the constructed support would have positive $2d/p$-Hausdorff measure or the limit measure's Fourier transform would not lie in $L^p$.

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Extended reading notes

Core claim

Theorem 1.6 asserts that for every $2<p<\infty$ there is a compact $S\subset\mathbb{R}^d$ and a probability measure $\mu$ with $\operatorname{supp}\mu\subset S$, $\hat{\mu}\in L^p(\mathbb{R}^d)$, and $H^{2d/p}(S)=0$. The proof builds $S$ and $\mu$ simultaneously: a weighted tree encodes a Cantor-type set whose cubes shrink at the rate $r_k=M_k^{-p/(2d)}b(Q_k)^{p/(2d)}/(n(Q_k)+1)$, and a random measure in each cube replaces the uniform Lebesgue measure by many small random shifts. Passing to a weak* limit along a rapidly growing sequence of branching numbers gives $\mu_\infty$, whose Fourier transform is shown to lie in $L^p$ via a layer-wise estimate; the side-length exponent is engineered so that the same choice makes the Hausdorff covering sum vanish and the $L^p$ norm increments summable. As a corollary, Raani's theorem (which uses packing measure) cannot be extended to Hausdorff measures.

Load-bearing premise

The load-bearing prerequisite is the exact side-length relation (2.5), $l(Q_j)=M_k^{-p/(2d)}b(Q_k)^{p/(2d)}/(n(Q_k)+1)$, together with the implicit requirement that every branching number $M_k$ is at least $2$; if the exponent $p/(2d)$ is altered or a branch stops branching, the Hausdorff covering estimate (2.7) or the $L^p$ convergence sum (3.18) fails.

Editorial extensions

If this is right

  • The endpoint uncertainty principle (Theorem 1.1 at $\alpha=2d/p$) is false for arbitrary compact sets: finiteness of the critical Hausdorff measure no longer forces a measure with $L^p$ Fourier transform to vanish.
  • Theorem 1.5 cannot be generalised from packing measure to Hausdorff measure, since the constructed $S$ has $H^{2d/p}(S)=0$ yet supports such a measure.
  • Structural assumptions on the support are essential: smooth surfaces satisfy the endpoint uniqueness (Theorems 1.3 and 1.4), while the irregular Cantor-type sets constructed here do not.
  • The construction is sensitive to Lorentz-space refinements of $L^p$; the author states that an exact version of the uncertainty principle for Lorentz spaces and Netrusov–Hausdorff capacities will appear in a separate article.

Reading between the lines

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

  • One plausible reading is that the critical dimension for the uncertainty principle is governed by a capacity- or packing-type quantity rather than by Hausdorff measure itself; the paper's method tests exactly the boundary where these two notions diverge.
  • A direct numerical experiment could check the mechanism: in dimension $d=1$ with $p=3$ and a fast growing sequence such as $M_k=2^{2^k}$, approximating $\mu_k$ on a grid should show bounded $L^3$ Fourier norms while the computed $2/3$-dimensional Hausdorff measure of the support tends to zero.
  • The probabilistic selection of the measure could likely be derandomized by averaging over shifts, turning the existence proof into an explicit deterministic construction and making the dependence on the sequence $M_k$ more transparent.
  • If the side-length exponent in (2.5) is changed even slightly, either the Hausdorff measure estimate or the $L^p$ summability fails; this suggests the measure sits exactly at the threshold and that refined Lorentz-space estimates will exhibit the same critical behaviour.
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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

1 major / 5 minor

Summary. The paper claims the following: for every 2<p<∞ there is a compact set S⊂R^d with H^{2d/p}(S)=0 and a probability measure μ supported on S such that \hat μ∈L^p(R^d). The proof builds a random Cantor-type set with a tree structure in which each vertex Q_k receives M_k children of side length r_k = M_k^{-p/(2d)} b(Q_k)^{p/(2d)}/(n(Q_k)+1), where b(Q_k) is the product of the inverse branching numbers along the path to the root. The measure μ is the weak-* limit of measures μ_k that replace the cube of Q_k by the averaged sum of the cubes of its children. The key estimates show that the Hausdorff covering sum at level n is n^{-2d/p} and that the L^p norm of \hat μ_k increases by at most C b(Q_k)^{p/2}(n(Q_k)+1)^d at each step; the latter sum converges because b(Q_k)≤2^{-n} and p>2. A probabilistic selection of the child configurations yields the required L^p and L^{p1} estimates at every stage.

Significance. If the proof is correct, the result settles the limit case α=2d/p of the uncertainty principle in a negative direction: H^{2d/p}(S)<∞ does not force a measure with \hat μ∈L^p and supp μ⊂S to vanish. This complements the positive results for packing measures (Theorem 1.5) and shows that the Hausdorff-measure analogue fails. The construction is explicit, self-contained, and the main estimates are checkable; the random Cantor construction with branch-dependent scaling is a useful technique. The main weaknesses are presentation issues in the definitions of μ_k and a lemma statement that is broader than its proof; these are local and do not affect the central argument once corrected.

major comments (1)
  1. [Section 2, Eqs. (2.8)–(2.10) and (2.15)] The measures μ_k are defined in two incompatible ways. If (2.8) is read literally, then μ_0 is the sum over the first-generation cubes weighted by b(Q_i), not λ_{[0,1]^d}; (2.9) asserts μ_0=λ_{[0,1]^d}. The recurrence (2.10) and its random version (2.15) are only consistent with (2.8) when μ_0 is taken as the measure after the first branching, i.e., when one sets μ_{-1}=λ_{[0,1]^d} and defines μ_k for k≥0 by (2.8). As printed, the induction in Lemma 3.6 and the estimate (3.18) do not parse. Please repair the indexing by introducing μ_{-1} and deleting or correcting (2.9).
minor comments (5)
  1. [Lemma 3.2] Lemma 3.2 is stated for all p>1, but its proof uses Lemma 4.1, which is valid only for p>2. Since the paper only applies Lemma 3.2 with p>2 and p1>p, the statement should be restricted to p>2 (or the proof extended to 1<p≤2).
  2. [Eq. (2.5)] The formula for r_k should be displayed as a fraction: r_k = M_k^{-p/(2d)} b(Q_k)^{p/(2d)}/(n(Q_k)+1). The current typography is ambiguous (it appears as multiplication by n+1), which would break the equalities in (2.7) and (3.15).
  3. [Section 3, Eq. (3.18)] The proof of (3.18) uses the bound b(Q_j)≤2^{-n(Q_j)}, which holds only if all branching numbers along the path are at least 2. This hypothesis should be stated explicitly when choosing the sequence M_k. Similarly, the condition 0<r<1/2 in Definition 2.3 requires r_k/l(Q_k)<1/2 for every k; this should be ensured by taking M_k sufficiently large.
  4. [Section 3, last paragraph] The passage to the weak-* limit should be spelled out: take a subsequence μ_{k_l} converging weak-* to μ_∞; then \hat μ_{k_l}→\hat μ_∞ pointwise, and Fatou's lemma gives \hat μ_∞∈L^p from the uniform bound. As written, the existence of the limit and the L^p conclusion are implicit.
  5. [Throughout] There are several typographical issues: the title has "Introducion"; Section 2 has "concrusted" and "paren"; Section 3 has "grouth"; the references contain "abelean". These do not affect the mathematics but should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is self-contained and all estimates are proven from stated definitions.

full rationale

Theorem 1.6 is an existence theorem proved by an explicit random Cantor-type construction. The cube side-length relation (2.5) is chosen as part of the construction, and every later estimate uses it directly rather than assuming the conclusion. Proposition 2.1 obtains the Hausdorff covering sum equal to n^{-2d/p} by substituting (2.5) into the cube sizes, which is a direct calculation, not a fitted prediction. The L^p bound (3.18) is obtained by summing the proven increment estimates from Lemma 3.6 and Corollary 3.7; those lemmas are proved within the paper from the randomization estimates of Lemma 3.2 and Lemma 3.4, and the auxiliary probabilistic inequality Lemma 4.1 is cited to the standard Marcinkiewicz-Zygmund inequality and proved by Hölder. Lemma 4.2 is also proved in the paper. The weak-* limit step is standard: µ_k are probability measures on a compact set, µ_k → µ_∞ weak-* implies the Fourier transforms converge pointwise, and Fatou gives ˆµ_∞ ∈ L^p from the uniform bounds. No fitted parameter is later renamed as a prediction, and no load-bearing self-citation occurs: the references to prior work are contextual results on the uncertainty principle or standard inequalities, while the central construction and estimates are self-contained. The only issues are minor presentation gaps, such as not explicitly stating that the rapidly growing M_k should be chosen ≥ 2 for the b(Q_j) ≤ 2^{-n} bound used in (3.18), and Lemma 3.2 being stated for p > 1 while its proof uses Lemma 4.1, which requires p > 2; these do not affect the p > 2 regime used in Theorem 1.6. Hence the derivation is not circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The construction introduces a random tree and cube sequence but no physically new entities; the free parameters are honest degrees of freedom in the construction.

free parameters (3)
  • M_k (branching numbers) = not specified; chosen sufficiently large (rapidly growing)
    The proof requires each M_k to be large enough for Corollary 3.7 and the L^{p1} decay (3.14) to hold; the values are free within these constraints.
  • p1 = any p1 > p
    Introduced in Corollary 3.3 to get extra decay in L^{p1}, used in Lemma 4.2 to control the L^p limit.
  • r_k (children side length) = r_k = M_k^{-p/(2d)} b(Q_k)^{p/(2d)}/(n(Q_k)+1)
    Defined in (2.5); the exact exponent p/(2d) is load-bearing for both the Hausdorff cover and the L^p summability.
assumptions (4)
  • standard math Marcinkiewicz-Zygmund inequality for independent zero-mean random variables
    Used in Lemma 4.1 to bound moments of sums; for p>2 as applied.
  • standard math Hausdorff-Young inequality
    Used in Lemma 3.4 to transfer L^{p'} control of measures to L^p control of Fourier transforms.
  • standard math Weak* compactness and Fatou's lemma for subsequential limits of measures
    Used to pass from uniformly bounded \hat μ_k in L^p to a limit μ_∞ with \hat μ_∞ ∈ L^p.
  • ad hoc to paper All M_k ≥ 2 along every root-to-leaf path
    Implicitly used for b(Q_k) ≤ 2^{-n} in (3.18); not stated in the construction. Easily satisfied by the 'rapidly growing' choice.

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Cite this review

Pith. "Pith review of A measure with small support and p-summable Fourier transform." pith.science (2026). https://pith.science/paper/VWXOXEUP

@misc{pith2026241207314,
  author       = {Pith},
  title        = {Pith review of: A measure with small support and p-summable Fourier transform},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VWXOXEUP}},
  note         = {Machine review of arXiv:2412.07314}
}
abstract

We construct a probability measure $\mu$ supported on a set of zero $2d/p$-Hausdorff measure such that $\hat{\mu}\in L_{p}(\mathbb{R}^d)$.

Figures

Figures reproduced from arXiv: 2412.07314 by the authors.

Figure 1
Figure 1. Tree T2 for M0 = 3, M1 = 4, M2 = 6. We define the weight of the vertex Qi by the formula b(Qi) = Y Qj is an ancestor of Qi M−1 j . (2.3) We say that Qk belongs to the n th layer if it has exactly n ancestors. We have bound b(Qk) ⩽ 2 −n, if Qk belongs to the n th layer. The notation n(Qk) means the number of the layer of Qk. Clearly, X n(Qk)=n b(Qk) = 1. (2.4) Now let Qk (the vertices) be cubes in R d and let l(Q) th… view at source ↗
Figure 2
Figure 2. Graph of fr for r = 0.1 and µˆM,r(x) D= 1 M X M j=1 e −2πi<βr,j ,x>λˆ 0(xr). (2.14) Here λ0 is the Lebesgue measure on [0, 1]d and {βr,j}M j=1 is the sequence of independent vectors that are uniformly distributed on the cube [0, 1 − r] d . The notation D= means equality of distributions. Let νM,r = µM,r(ω) be the value of µM,r at some point ω of the probability space to be chosen later. By definitions, νM,r is a pro… view at source ↗
Figure 3
Figure 3. Construction 3 Estimates Lemma 3.1. For all M ∈ N, r < 1 2 and p > 1 the inequality Z Rd |EµˆM,r(x)| p dx ≲ 1 (3.1) is true. The constant depends only on d and p. Proof. By (2.14), we have EµˆM,r(x) = E 1 M X M j=1 e −2πi<βr,j ,x>λˆ 0(rx) = λˆ 0(rx)Ee −2πi<βr,1,x>. (3.2) Since λ0 is a probability measure, |λˆ 0| ⩽ 1. With the notation h(t) = 1−e it t , we have |EµˆM,r(x)| = [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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

  1. $L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve

    math.CA 2024-12 conditional novelty 7.0 of 10

    If a function lies in L^p and its Fourier transform is supported on a small fractal set on the moment curve, then for p up to a sharp threshold (d^2+d+2α)/(2α) for d≥3, and 4/α for d=2, the function is identically zero.

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