{"id":"6a4960db-1af2-429c-8945-92394171d513","arxiv_id":"2412.07314","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"For every 2<p<∞, a probability measure on R^d exists with \\hat μ ∈ L^p supported on a compact set of zero H^{2d/p} measure.","lead":"This paper constructs a probability measure whose Fourier transform is p-integrable even though the measure sits on a set with zero (2d/p)-dimensional Hausdorff measure. It closes the endpoint case of a classical uncertainty principle: the expected vanishing result is false at the critical exponent.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the construction's key estimates appear internally consistent, and the only issues are minor, fixable presentation gaps.","rationale":"The reader's weakest assumption identifies the side-length relation (2.5) as the hinge of the proof, and I agree that this relation is load-bearing: all later estimates depend on its exact exponent. However, on independent re-derivation the relation is internally consistent and delivers exactly the two convergences claimed: the Hausdorff covering sum in Proposition 2.1 and the L^p increment sum in (3.18). The implicit M_k ≥ 2 condition flagged by the reader is real but not a threat to the theorem, since the sequence can be chosen to satisfy it from the start. Lemma 3.2's p>1 statement is a typo in the hypothesis; the proof and all uses require p>2, which holds in Theorem 1.6. The weak-* limit argument is not written out in full, but it is standard and does not create a gap. I therefore do not find a load-bearing objection, and the reader's conditional verdict with minor requested corrections remains appropriate.","tokens_in":7565,"tokens_out":33838,"duration_ms":334042,"concrete_test":"Independently re-derive (3.15) from Corollary 3.5 and (2.5), then repeat the derivation with the exponent p/(2d) in (2.5) replaced by p/(2d)+ε for a small nonzero ε. Verify that for ε≠0 either the Hausdorff covering sum in (2.7) no longer tends to 0 or the L^p sum in (3.18) diverges, confirming that the exact exponent in (2.5) is load-bearing but correctly chosen. Also confirm that choosing all M_k ≥ 2 makes the bound b(Q_j) ≤ 2^{-n} hold along every root-to-leaf path.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the two critical estimate chains that support Theorem 1.6. The side-length relation (2.5) is exactly what makes the Hausdorff covering sum in Proposition 2.1 equal to n^{-2d/p}, and it makes the L^p increment in Lemma 3.6 reduce to b(Q_k)^{p/2}(n(Q_k)+1)^d, so the telescoping sum in (3.18) converges for p>2. The exponent balance is internally consistent: the first term in (3.15) becomes independent of M_k precisely because r_k^{-d} = M_k^{p/2} b(Q_k)^{-p/2}(n+1)^d, and the second term decays because p/p'=p-1>1. The weak-* limit step is standard: μ_k are probability measures on a compact set, μ_k → μ_∞ weak-* implies ̂μ_k → ̂μ_∞ pointwise, and Fatou gives ̂μ_∞ ∈ L^p from the uniform bound. The remaining issues are presentation-level: Lemma 3.2 states p>1 while its proof uses Lemma 4.1 which requires p>2, but the theorem only uses p>2 and p1>p, so this is harmless; and the proof never explicitly states that all M_k must be chosen ≥ 2 for the bound b(Q_j) ≤ 2^{-n} used in (3.18). Choosing the sequence M_k to be rapidly growing with M_k ≥ 2 fixes this. Neither issue threatens the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":7893,"tokens_out":36511,"duration_ms":344801,"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":[{"comment":"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).","section":"Section 2, Eqs. (2.8)–(2.10) and (2.15)"}],"minor_comments":[{"comment":"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).","section":"Lemma 3.2"},{"comment":"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).","section":"Eq. (2.5)"},{"comment":"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.","section":"Section 3, Eq. (3.18)"},{"comment":"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.","section":"Section 3, last paragraph"},{"comment":"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.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript presents a clean counterexample at the endpoint of the uncertainty principle and is likely to be of interest to readers in harmonic analysis and geometric measure theory. The formal issues are concentrated in the definition of μ_k and the statement of Lemma 3.2; both are fixable locally and do not affect the central construction. I recommend a minor revision rather than rejection. The paper is appropriate for a journal in analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this paper settles the endpoint case of the Hausdorff-measure uncertainty principle in the negative. For any 2<p<∞, the author constructs a probability measure μ supported on a compact set S with zero 2d/p-dimensional Hausdorff measure and with Fourier transform in L^p. That kills the natural endpoint extension of Theorem 1.1 (the Salem/Edgar-Rosenblatt/Kahane line). The construction is a branch-dependent random Cantor set: the number of children M_k and side lengths r_k vary along the tree, which is a genuine twist on Salem's and Bluhm's random sets and is what lets L^p and L^{p_1} behavior separate. That is real novelty, not repackaging.\n\nWhat the paper does well: the proof is fully self-contained and the main estimate chain is sound. I checked the two critical places. The side-length relation (2.5) is chosen exactly so the Hausdorff covering sum in Proposition 2.1 becomes n^{-2d/p}, and the L^p increment in Lemma 3.6 reduces to b(Q_k)^{p/2}(n+1)^d, which sums over the tree because of the weight bound b(Q_k) ≤ 2^{-n}. The telescoping argument in (3.18) converges for p>2. The weak-* limit step is standard, and Fatou gives the L^p bound. I see no circularity; the M_k are chosen ex post to force convergence, which is legal in an existence proof.\n\nSoft spots, in order of real weight: (1) Lemma 3.2 is stated for p>1 but the proof invokes Lemma 4.1, which requires p>2. The theorem only needs p>2, so the fix is to restate Lemma 3.2 as p>2. Misstated but harmless to the main result. (2) The definition of μ_k in (2.8) and the recurrence (2.10) have an indexing inconsistency: μ_0 is defined twice, once as a sum over leaves and once as λ_{[0,1]^d}; the latter is intended, and the sum in (2.8) should be read accordingly. Read charitably it is clear enough, but it will confuse a careful reader. (3) The proof never states the standing assumption that every M_k ≥ 2 along every root-to-leaf path. The bound b(Q_j) ≤ 2^{-n} in (3.18) needs it. Choosing M_k rapidly increasing with M_k ≥ 2 fixes this, and the author does say \"rapidly growing\" but should make the lower bound explicit. These are presentation-level and do not threaten the central theorem.\n\nWho this is for: harmonic analysts and geometric measure theorists working on Fourier dimension and uncertainty. It deserves a serious referee; with the small corrections it is a clean, publishable note. I'd send it out.","headline":"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.","tokens_in":8369,"tokens_out":1681,"would_cite":true,"duration_ms":16928,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B10","28A78","28A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["uncertainty principle","Fourier transform","Hausdorff measure","Cantor-type set","p-summable Fourier transform","L^p integrability","random construction","packing measure"],"falsifier":"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 $L^p$.","tokens_in":7363,"feed_emoji":"📐","tokens_out":14381,"duration_ms":136962,"temperature":0.7,"pith_summary":"This paper proves that the endpoint form of the classical uncertainty principle fails for general compact sets. For any $2<p<\\infty$, the author constructs a probability measure $\\mu$ whose Fourier transform lies in $L^p(\\mathbb{R}^d)$ while its support is a compact set $S$ with zero $2d/p$-dimensional Hausdorff measure. That is the exact boundary case $\\alpha=2d/p$ of Theorem 1.1, where previously only uniqueness results under extra assumptions (smooth surfaces, finite packing measure) were known. The construction is an explicit random Cantor-type set with branch-dependent shrinking rates, chosen so that the Hausdorff covering sums and the $L^p$ norm estimates balance precisely.","feed_headline":"Zero-Hausdorff support can still host an Lp Fourier transform","feed_subtitle":"A random Cantor construction disproves the endpoint uncertainty principle for general compact sets.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Salem's one-dimensional result shows the endpoint cannot be improved there, motivating the critical exponent question.","marker":"[10]"},{"why":"Edgar and Rosenblatt proved Theorem 1.1 in the case $d-1\\le\\alpha$, the classical uncertainty principle this paper refutes at the endpoint.","marker":"[6]"},{"why":"Agranovskiy and Narayanan proved endpoint uniqueness for $C^1$ surfaces, a structural assumption that prevents the phenomenon constructed here.","marker":"[2]"},{"why":"Raani's packing-measure endpoint theorem is the result shown here to be impossible to generalise to Hausdorff measures.","marker":"[11]"},{"why":"Rosenblatt's smooth-surface endpoint theorem provides another regularity assumption that the new construction avoids.","marker":"[13]"},{"why":"Bluhm's random recursive Salem sets supply the random Cantor construction technique adapted to the variable-branching setting.","marker":"[4]"}],"fun_headline_variants":["Zero-Hausdorff support with Lp Fourier: a measure exists","Tiny support, p-summable Fourier: measure defies Hausdorff limit","Defying endpoint uncertainty: zero-Hausdorff set holds Lp Fourier","Small set, Lp Fourier transform: a counterexample to Hausdorff","Cantor-style measure: zero-Hausdorff support, Lp Fourier"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Zero-Hausdorff support with Lp Fourier: a measure exists","Tiny support, p-summable Fourier: measure defies Hausdorff limit","Defying endpoint uncertainty: zero-Hausdorff set holds Lp Fourier","Small set, Lp Fourier transform: a counterexample to Hausdorff","Cantor-style measure: zero-Hausdorff support, Lp Fourier"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001129,"raw_usage":{"total_tokens":4613,"prompt_tokens":787,"completion_tokens":3826,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":403,"completion_tokens_details":{"reasoning_tokens":3724}},"tokens_in":403,"tokens_out":3826,"duration_ms":26645,"temperature":1.0,"reasoning_tokens":3724,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:57:06.574122+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $L^p$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Edgar and Rosenblatt proved Theorem 1.1 in the case $d-1\\le\\alpha$, the classical uncertainty principle this paper refutes at the endpoint."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Rosenblatt's smooth-surface endpoint theorem provides another regularity assumption that the new construction avoids."},{"cited_title":"Bluhm,Random recursive construction of Salem sets, Ark","cited_arxiv_id":null,"evidence_quote":"Bluhm's random recursive Salem sets supply the random Cantor construction technique adapted to the variable-branching setting."}],"review_version":1}