{"id":"a2a28374-cb99-41e1-8a71-1fe4d6fd8e7f","arxiv_id":"2508.20746","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The zero set of the Gaussian entire function yields local sampling inequalities for Fock spaces, allowing polynomial sampling with d+o(d) points and sub-polynomial constants.","lead":"This paper proves quantitative sampling inequalities for the zero set of the Gaussian entire function on Fock spaces, using a new weighted version of Seip and Wallsten's sampling theorem. The authors show that d+o(d) zeroes can sample polynomials of degree d with sampling constants that grow slower than d^epsilon.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.4's use of [28] is misstated (log^* P instead of log P, inconsistent additive error), so Lemma 6.1(II) is not currently established; the correction must be checked before the main theorem is fully rigorous.","rationale":"I read the paper as a two-stage argument: a deterministic weighted sampling theorem (Proposition 4.2) plus a probabilistic verification that the GEF zeros satisfy its hypotheses (Lemma 6.1). The reader's weakest assumption was the external perturbed-lattice tail (15). That is a legitimate sensitivity point, but it is a cited external input and is used with a comfortable margin. The more immediate problem is internal: Proposition 5.4, on which Lemma 6.1(II) relies, contains a misquoted bound from [28] and an inconsistent additive error term. The paper's own proof of Lemma 6.1(II) therefore does not go through as written. This is not a stylistic issue, because without a valid Proposition 5.4 the separation condition (ii) is unproved and Theorem 1.1 has no support. The fix is likely straightforward—the [28] theorem certainly gives an exponentially small error—but it must be written out and checked with the exact constants in the regime N=T^6, \\lambda=T^4, where the present text is vacuous. Hence the conditional verdict is appropriate; no change of verdict is needed, but the paper should not be accepted as fully rigorous until this is corrected.","tokens_in":33118,"tokens_out":34660,"duration_ms":333115,"concrete_test":"Re-derive the approximation step using the exact statement of [28, Theorem 3.1]: set N=T^6(R'), \\tau=T(R'), \\varsigma=1/\\log^*R', and compute the probability that max_j max_{\\tilde Q_j}|G_j^*| \\ge e^{-\\tau^2}. Then redo the proof of Proposition 5.4 with the corrected tail P \\le \\exp(C + \\log^* N - e^{\\tau^2}) and the resulting additive term. Check whether the final bound is \\le e^{-c T^4(R')} + \\exp(-e^{c' T^2(R')}) (or at least \\le \\exp(-c\\log^*R'\\,\\log^*\\log^*R')) uniformly in R'. If the corrected term does not yield a negligible probability, Lemma 6.1(II) fails and the proof of Theorem 1.1 must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the large-scale separation estimate Proposition 5.4 contains a concrete internal gap. It quotes [28, Theorem 3.1] as log^* P(max_j max_{\\tilde Q_j} |G_j^*| \\ge e^{-\\tau^2}) \\le C + \\log^* N - e^{\\tau^2}. Since log^* x = max(1, log x), for a probability this left side is identically 1 (or, for very small P, still 1 under the max convention), so the displayed inequality is false for large \\tau. The intended statement must use log P or give a tail for P directly. The additive term displayed after (34), written e^{C N e^{-e^{\\tau^2}}}, is inconsistent with such a corrected tail: with N = T^6(R') and \\tau = T(R') it is approximately 1, which makes the bound vacuous. Lemma 6.1(II) applies Proposition 5.4 to N=T^6(R') squares and needs failure probability e^{-c T^4(R')} plus a negligible term; as written, this step does not go through. This is load-bearing because the separation condition (ii) is one of the two hypotheses of Proposition 4.2 needed for Theorems 1.1 and 1.6.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies sampling inequalities for the Fock space using the zero set of the Gaussian entire function (GEF). Its main results are: (i) a quantitative, weighted relaxation of Seip--Wallstén's sampling theorem (Proposition 4.2) that permits local density and separation defects; (ii) probabilistic estimates on the separation of GEF zeroes (Lemma 5.2, Propositions 5.3 and 5.4); (iii) Theorem 1.1, a weighted sampling inequality over all zeroes holding with high probability, with a slowly growing weight; and (iv) Theorem 1.6, which gives sampling inequalities for polynomials of degree at most d using d+o(d) zeroes, with sampling constants growing slower than d^ε for any ε>0. The paper also recovers the Lyons--Zhai uniqueness theorem for the GEF. The proof strategy is to show that the GEF zeroes satisfy the hypotheses of the deterministic Proposition 4.2 with high probability, using the perturbed lattice description of Sodin--Tsirelson and large-deviation estimates for zeroes.","tokens_in":33402,"tokens_out":15495,"duration_ms":155129,"significance":"If the technical gap discussed below is repaired, this is a valuable contribution. The paper supplies quantitative sampling bounds for a natural repulsive random point process near the critical density, with polynomial sampling constants subpolynomial in the degree, and its deterministic framework (Proposition 4.2) is likely to be reusable. The proofs are generally detailed, with explicit probability bounds and constant tracking, and the argument is not circular: it builds on external theorems (Seip--Wallstén, Sodin--Tsirelson, Nazarov--Sodin, Krishnapur) rather than assuming the target inequalities. The main probabilistic separation estimates are also of independent interest. However, one load-bearing step, Proposition 5.4, currently contains a misquoted external estimate, and as written the proof of Lemma 6.1(II) does not go through.","major_comments":[{"comment":"The estimate quoted from [28, Theorem 3.1] is stated as log^* P(max_j max_{z∈eQ_j}|G_j^*(z)| ≥ e^{-τ^2}) ≤ C + log^* N - e^{τ^2}. With the paper's definition log^* x = max(1, log x), the left-hand side is identically 1 for any probability P, so the displayed inequality is false for large τ. The intended statement must involve log P (or a direct tail bound). The additive term displayed after (34), e^{CN e^{-e^{Lτ^2}}}, is also inconsistent with such a corrected tail: in the application in Lemma 6.1(II), where N=T^6(R') and τ=T(R'), this term is approximately 1, so it does not provide the needed negligible failure probability. Since Lemma 6.1(II) is the step verifying condition (ii) of Proposition 4.2, Theorem 1.1 and Theorem 1.6 are not fully established as written. The gap appears localized and repairable: with the correct tail from [28], the additive term should become C N e^{-e^{Lτ^2}}","section":"Proposition 5.4 and Lemma 6.1(II)"},{"comment":"The application of Proposition 5.4 in the proof of Lemma 6.1(II) with τ=T(R'), N=T^6(R'), ς=(log^*R')^{-1} and λ=T^4(R') is asserted to satisfy the hypotheses 'provided R' is sufficiently large', but the verification is not shown. This is a routine but necessary check: after correcting Proposition 5.4, the authors should explicitly confirm that λ dominates the threshold D N(L^3τ^2ς^4 + Lτ^3ς^{-1}e^{-Lτ^2}) and that the corrected additive term is indeed bounded by the negligible probability claimed. This is not a conceptual obstacle, but it is needed for the rigorous conclusion.","section":"Lemma 6.1(II) parameter verification"}],"minor_comments":[{"comment":"The paper defines log^* x = max(1, log x), but the quoted estimate from [28] appears to use a different convention (likely log). Please clarify the convention used in the citation and ensure consistency throughout.","section":"Section 2.1"},{"comment":"The verification that the translated sets Z_{kl} satisfy conditions (i) and (ii), and the final summation estimate (27), are delegated to 'straightforward but tedious' computations. Since the constant tracking is central to the L→1 dependence, please expand these computations or provide the key intermediate estimates.","section":"Proposition 4.2 proof"},{"comment":"The optimality discussion is presented as a proof sketch, with the hole construction 'skipped'. Since this remark is not needed for the main theorems, it should be explicitly labeled as a sketch or moved to a 'conjecture/remark' status.","section":"Remark 6.3"},{"comment":"There are several typos: Remark 3.1 'slighltly'; Section 2.3 'purpuses'; Lemma 6.1 proof 'stament' and an extraneous 'oo' in equation (37). These should be corrected.","section":"Typos"}],"recommendation":"major_revision","confidential_remarks":"The paper is promising and the central strategy appears sound, but the misquotation of [28] in Proposition 5.4 is a genuine load-bearing error. I recommend major revision: the authors should correct the statement and proof of Proposition 5.4, verify the subsequent parameter choices in Lemma 6.1(II), and re-check the final sampling theorems. I do not see evidence of circularity or fitting; the issue is a technical gap that should be fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth a serious referee. The main contribution is Proposition 4.2, a quantitative weighted version of Seip-Wallsten that tolerates bad points and local separation conditions, plus the probabilistic machinery to show the GEF zeroes satisfy those conditions with high probability. The resulting Theorems 1.1 and 1.6 are genuinely new: d+o(d) points sample polynomials of degree d with sub-polynomial constants, and Lyons-Zhai's uniqueness result is recovered as a corollary. The paper is honest about what is new and what is a recovery, and the citation pattern looks sound.\n\nThe weak spot is exactly where the stress-test note points. Proposition 5.4 quotes [28, Theorem 3.1] as\nlog^* P(max_j max_{eQ_j} |G*_j| \\ge e^{-\\tau^2}) \\le C + \\log^* N - e^{\\tau^2}.\nBut log^* x = max(1, log x), so for a probability the left side is identically 1, and the inequality is false for large \\tau. The intended statement must use log P, giving a tail like P \\le e^C N e^{-e^{\\tau^2}}. The additive term in (34) is also written e^{C N e^{-e^{\\tau^2}}}, which is about 1 for the parameters used in Lemma 6.1(II), making the bound vacuous. Since Lemma 6.1(II) feeds condition (ii) of Proposition 4.2, this is load-bearing. The fix looks straightforward—the corrected tail from [28] should make the argument work—but as written the separation estimate is not established.\n\nThe other compressed steps ('straightforward but tedious') are acceptable for this area, and the perturbed lattice tail (15) is a legitimate external input. No circularity, no parameter fitting.\n\nFor a reader working on random sampling or Fock spaces, this is a valuable paper to engage with. It deserves a serious referee, and the referee should ask the authors to correct Proposition 5.4 before acceptance.","headline":"Strong quantitative sampling results for GEF zeroes, but a misstatement in Proposition 5.4 needs fixing.","tokens_in":33915,"tokens_out":4949,"would_cite":true,"duration_ms":47777,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["30H20","94A20","60G15","30C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The zero set of the Gaussian entire function yields weighted sampling inequalities on Fock spaces and samples degree-d polynomials from d+o(d) points with sub-polynomial sampling constants.","keywords":["Gaussian entire function","zero set","Fock space","random sampling","sampling constants","perturbed lattice","Marcinkiewicz–Zygmund inequalities","Seip–Wallstén density"],"falsifier":"Simulate the Gaussian entire function with intensity L_d = 1 + C/log* log* d, take the zeroes in a disk of radius R_d with R_d^2 = d + C sqrt(d log* d), and compute the best sampling constants A_d, B_d for degree-d polynomials. The theorem predicts about d+o(d) points and B_d/A_d sub-polynomial in d; if the minimal sample count exceeds d + C sqrt(d log* d) or the constant ratio grows like d^alpha with alpha>0 for d up to a few thousand, the claimed sampling bound is wrong. Alternatively, empirically check the perturbed-lattice tail (15): if deviations tau occur with probability decaying slower","tokens_in":32992,"feed_emoji":"🎲","tokens_out":12024,"duration_ms":112662,"temperature":0.7,"pith_summary":"The paper proves that the random zero set of the Gaussian entire function—the natural translation-invariant random analytic function—carries almost enough information to reconstruct Fock-space functions from point samples. The main result is a weighted sampling inequality: every Fock function's L^p norm is bounded by a weighted sum of its values on the zero set, with a slowly growing weight that is near-optimal. For polynomials of degree d, the same zero set restricted to a disk of radius about sqrt(d) provides sampling inequalities using d+o(d) points, and with high probability the sampling constants grow more slowly than d^epsilon for any epsilon>0. This gives a quantitative form of the known uniqueness result for the Fock space and a first step toward sampling from repulsive random point processes.","feed_headline":"d+o(d) random zeroes are enough to sample degree-d polynomials","feed_subtitle":"Sampling constants stay sub-polynomial in d, so no oversampling beyond the critical density is needed.","key_machinery":"Three mechanisms carry the argument. (1) The perturbed-lattice description of the zeroes: Z(L) is statistically the lattice sqrt(pi/L) Z^2 plus perturbations with tail P(sqrt(L)|xi| > tau) <= C exp(-c tau^4/log* tau); this connects the point process to the deterministic lattice theory. (2) A quantitative Seip-Wallsten theorem (Proposition 4.2) that relaxes uniform separation to a product-separation condition S_Z(z) = product over nearby distinct points w of |z-w|, allowing a small proportion of badly separated points and producing a weight omega that depends on a tolerance function T. (3) Separation estimates for the zeroes: a small-scale bound (Proposition 5.3) comparing k-point intensities","core_discovery":"On the paper's own terms, the central discovery is that the zero set Z(L) of the Gaussian entire function satisfies a quantitative weighted sampling inequality for the whole Fock space F^p (Theorem 1.1): for any L>1 and 1<=p<infinity, outside an event of probability at most exp(-c log* R log* log* R), every f in F^p obeys ||f||_p^p <= C' sum_{z in Z(L)} omega(|z|+R) |f(z)|^p e^{-p|z|^2/2}, where omega grows like exp(C (log* r)^{1/2} (log* log* r)^6). For polynomials of degree at most d, choosing L_d = 1 + C/log* log* d and taking the zeroes in a disk of radius R_d with R_d^2 = d + C sqrt(d log* d), about d+o(d) points suffice and both sampling constants grow slower than d^epsilon for every e","pith_inferences":["The same deterministic framework should transfer to any repulsive point process that has a perturbed-lattice description with comparable tail bounds, such as the Ginibre ensemble, suggesting that d+o(d) sampling points could be achievable there too; the paper states this only as motivation.","Because the weight omega(r) in Theorem 1.1 is essentially optimal, the sample count in Theorem 1.6 cannot be pushed below d - c sqrt(d log* d) without forcing the sampling constants to grow; the d+o(d) form is therefore the right order.","A numerical consequence: for moderate d, the empirical ratio of sampling constants for the zero set should track exp(C (log* d)^{1/2} (log* log* d)^6); measuring this would give an early check of the sharpness claims.","The Bessel-type upper bound for polynomials suggests a route to relevant sampling: functions concentrated on a disk of radius about sqrt(d) can be sampled from the same zero set with the same sub-polynomial constants, even without the full Fock-space inequality."],"forward_implications":["Stable reconstruction of Fock-space functions from samples at the zeroes of the Gaussian entire function is possible, with a weight that is essentially optimal up to log* log* factors in the exponent.","For polynomials of degree d, the minimal number of sampling points is attained up to o(d): one needs d+o(d) points, and the sampling constants are sub-polynomial in d.","The results recover, for the Gaussian entire function, the uniqueness theorem of Lyons and Zhai: almost surely the zero set is a uniqueness set for the Fock space (Corollary 1.3).","The quantitative estimates on how many zeroes can be found close together have independent value for studying local fluctuations of the zero process.","With a mild oversampling of slightly more than sqrt(log* d), a Marcinkiewicz-Zygmund inequality holds with uniformly bounded constants (Proposition 1.4)."],"supporting_citations":[{"why":"Seip-Wallsten sampling theorem for Fock spaces; the deterministic result being quantified and relaxed in Proposition 4.2.","marker":"[37]"},{"why":"Seip's density theorem characterising sampling sets; supplies the critical density 1/pi baseline.","marker":"[35]"},{"why":"Perturbed-lattice description of the zeroes of the Gaussian entire function; provides the structural input (15) used everywhere.","marker":"[41]"},{"why":"Krishnapur's tail and overcrowding estimates; refines the perturbed-lattice tail to (15) and bounds the number of points in unit balls.","marker":"[23]"},{"why":"Strong clustering estimate for the k-point correlation of zeroes; basis of the small-scale separation Proposition 5.3 via Lemma 5.2.","marker":"[29]"},{"why":"Almost independence of the Gaussian entire function over long distances; used in the large-scale separation Proposition 5.4.","marker":"[28]"},{"why":"Large fluctuation estimates and Rouché-type Lemma 8; controls points in balls and the large-scale separation step.","marker":"[31]"},{"why":"Hole probability decay; used in Proposition 1.4 and Remark 6.3 to show the weight omega is near-optimal.","marker":"[40]"},{"why":"Lyons-Zhai uniqueness theorem for zero sets of Gaussian analytic functions; recovered for the GEF in Corollary 1.3.","marker":"[26]"}],"fun_headline_variants":["d+o(d) zeros suffice for polynomial sampling","Sub-polynomial sampling constants from random zeros","Random zeros hit near-optimal sampling density for polynomials","d+o(d) random zeros: sampling with sub-polynomial constants","Zero set of Gaussian function samples polynomials optimally"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The entire result rests on the fast tail decay in the perturbed-lattice description of the zeroes—the probability of a deviation tau is about exp(-c tau^4/log* tau)—and if that decay were slower, the weight omega would blow up and the d+o(d) polynomial sampling bound would fail.","fun_headline_variants_meta":{"raw":{"variants":["d+o(d) zeros suffice for polynomial sampling","Sub-polynomial sampling constants from random zeros","Random zeros hit near-optimal sampling density for polynomials","d+o(d) random zeros: sampling with sub-polynomial constants","Zero set of Gaussian function samples polynomials optimally"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00085,"raw_usage":{"total_tokens":3537,"prompt_tokens":751,"completion_tokens":2786,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":2720}},"tokens_in":495,"tokens_out":2786,"duration_ms":23426,"temperature":1.0,"reasoning_tokens":2720,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:51:18.179093+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the Gaussian entire function with intensity L_d = 1 + C/log* log* d, take the zeroes in a disk of radius R_d with R_d^2 = d + C sqrt(d log* d), and compute the best sampling constants A_d, B_d for degree-d polynomials. The theorem predicts about d+o(d) points and B_d/A_d sub-polynomial in d; if the minimal sample count exceeds d + C sqrt(d log* d) or the constant ratio grows like d^alpha with alpha>0 for d up to a few thousand, the claimed sampling bound is wrong. Alternatively, empirically check the perturbed-lattice tail (15): if deviations tau occur with probability decaying slower","supporting_citations":[{"cited_title":"Seip and R","cited_arxiv_id":null,"evidence_quote":"Seip-Wallsten sampling theorem for Fock spaces; the deterministic result being quantified and relaxed in Proposition 4.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Seip's density theorem characterising sampling sets; supplies the critical density 1/pi baseline."},{"cited_title":"Sodin and B","cited_arxiv_id":null,"evidence_quote":"Perturbed-lattice description of the zeroes of the Gaussian entire function; provides the structural input (15) used everywhere."},{"cited_title":"Krishnapur","cited_arxiv_id":null,"evidence_quote":"Krishnapur's tail and overcrowding estimates; refines the perturbed-lattice tail to (15) and bounds the number of points in unit balls."},{"cited_title":"Nazarov and M","cited_arxiv_id":null,"evidence_quote":"Strong clustering estimate for the k-point correlation of zeroes; basis of the small-scale separation Proposition 5.3 via Lemma 5.2."},{"cited_title":"Nazarov and M","cited_arxiv_id":null,"evidence_quote":"Almost independence of the Gaussian entire function over long distances; used in the large-scale separation Proposition 5.4."},{"cited_title":"Nazarov, M","cited_arxiv_id":null,"evidence_quote":"Large fluctuation estimates and Rouché-type Lemma 8; controls points in balls and the large-scale separation step."},{"cited_title":"Sodin and B","cited_arxiv_id":null,"evidence_quote":"Hole probability decay; used in Proposition 1.4 and Remark 6.3 to show the weight omega is near-optimal."},{"cited_title":"Lyons and A","cited_arxiv_id":null,"evidence_quote":"Lyons-Zhai uniqueness theorem for zero sets of Gaussian analytic functions; recovered for the GEF in Corollary 1.3."}],"review_version":1}