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Sampling properties of the zeroes of the Gaussian entire function

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Pith's one-line read 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.

desk verdict Strong quantitative sampling results for GEF zeroes, but a misstatement in Proposition 5.4 needs fixing. read the letter →

arxiv 2508.20746 v1 pith:TXE2ERHF submitted 2025-08-28 math.PR math.FA

Jeremiah Buckley , Felipe Marceca , Joaquín Singer This is my paper · ORCID
classification math.PRmath.FA MSC 30H2094A2060G1530C15
keywords GaussianentirefunctionzerosetFockspacerandomsamplingconstantsperturbedlatticeMarcinkiewicz–ZygmundinequalitiesSeip–Wallsténdensity
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

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.

What carries the argument

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

What would settle it

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

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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).

Reading between the lines

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

  • 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.
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Editorial analysis

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

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Proposition 5.4 and Lemma 6.1(II)] 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}}
  2. [Lemma 6.1(II) parameter verification] 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.
minor comments (4)
  1. [Section 2.1] 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.
  2. [Proposition 4.2 proof] 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.
  3. [Remark 6.3] 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.
  4. [Typos] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the sampling inequalities are derived from external deterministic and probabilistic theorems plus verified GEF estimates; no fitted parameters or self-citation chains.

full rationale

The derivation is self-contained relative to external input. Theorem 1.1 is obtained by combining a deterministic quantitative sampling statement (Proposition 4.2, proved in Section 4 from Seip-Wallsten's sufficiency theorem and standard Fock-space estimates) with a probabilistic verification in Lemma 6.1 that the GEF zero set satisfies the two required hypotheses (perturbed lattice and separation). The perturbed lattice description and the tail estimate in (15) come from Sodin-Tsirelson and Krishnapur, not from the target theorem; the separation estimates in Section 5 are proved from the GEF correlation structure via Lemmas 5.2 and Propositions 5.3/5.4, whose inputs are [29], [28], [31], and [33]. Neither the deterministic proposition nor the probabilistic checks assume the desired inequality (4) or the polynomial inequalities of Theorems 1.1/1.6. The constants c, C, C' are existential bounds produced by the proof; no parameter is fitted to the final inequality. Theorem 1.6 follows from Theorem 1.1 plus a Bessel inequality (Proposition 6.2), and Corollary 1.3 is a consequence, not an input. There are no self-citations by the authors and no uniqueness theorem imported from same-author prior work. The skeptic's concern that Proposition 5.4 may misstate the [28, Theorem 3.1] tail estimate (log^* P versus log P) is a potential correctness issue about the application of an external theorem, not a circular reduction, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central results rely on established theorems from the sampling and random analytic functions literature, which are treated as inputs. No free numerical parameters are fitted to data, and no new entities are posited. The hand-chosen auxiliary functions T, S, and kappa in Section 6 are analytical devices, not fitted parameters, and the final claims hold for any sufficiently large constants.

assumptions (6)
  • domain assumption Seip-Wallsten characterization and sufficiency theorem for sampling sets in Fock spaces: a set is sampling iff it is a finite union of separated sets and contains a separated subset with lower Beurling-Landau density > 1/pi.
    Invoked in the introduction and used as the benchmark for Proposition 4.2 and in the proof of Proposition 1.4 via [37, Theorem 1.1].
  • domain assumption Sodin-Tsirelson perturbed lattice description of zeroes of the Gaussian entire function, with the refined tail estimate P(sqrt(L)|z_00| > tau) <= C exp(-c tau^4/log* tau) in equation (15).
    Section 2.5, equation (15). This is the main structural input used to verify condition (i) of Proposition 4.2 in Lemma 6.1(I).
  • domain assumption Nazarov-Sodin almost independence for the GEF over long distances, used to decompose the normalized GEF into independent copies plus a small error term with probability bound of the form exp(-e^{c tau^2}).
    Used in the proof of Proposition 5.4 for the large-scale separation estimate; without this coupling the large-scale part of Lemma 6.1 would fail.
  • domain assumption Sodin-Tsirelson and Krishnapur estimates for hole probability, overcrowding, and number of points in balls for the GEF ([40, Theorem 1], [40, Theorem 2], [31, Theorem 1], [23, Theorem 1]).
    Used in Proposition 5.3, Proposition 6.2, and the optimality discussion in Remark 6.3 to bound the number of points in balls and the probability of large empty regions.
  • domain assumption Nazarov-Sodin bound on the two-point correlation function of the GEF ([29, Theorem 1.1]) and the explicit quantitative version proved in Lemma 5.2.
    Used in Lemma 2.3 and Proposition 5.3 to estimate the probability of finding close pairs of zeroes.
  • standard math Standard Fock space facts: subharmonicity estimate (8), isometric Bargmann-Fock shifts, and concentration operator eigenvalues with Chernoff bound (9)-(10).
    These are standard background results used in Lemma 2.1 and throughout the proofs of the sampling inequalities.

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Pith. "Pith review of Sampling properties of the zeroes of the Gaussian entire function." pith.science (2026). https://pith.science/paper/TXE2ERHF

@misc{pith2026250820746,
  author       = {Pith},
  title        = {Pith review of: Sampling properties of the zeroes of the Gaussian entire function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TXE2ERHF}},
  note         = {Machine review of arXiv:2508.20746}
}
abstract

We study sampling properties of the zero set of the Gaussian entire function on Fock spaces. Firstly, we relax Seip and Wallst\'en's density and separation conditions for sampling sets on Fock spaces to obtain weighted inequalities for sets that are not necessarily sampling. On the probabilistic front, we estimate the number of zeroes of the Gaussian entire functions that are close to each other. We use these to prove random sampling inequalities for polynomials of degree at most $d$ using ${d}+o(d)$ points, and show that, with high probability, the sampling constants grow slower than $d^\varepsilon$ for any $\varepsilon>0$. In particular, we recover a result from Lyons and Zhai in the case of the Gaussian entire function, where it is shown that the zeroes are (almost surely) a uniqueness set for the Fock space.

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