Stealthy point processes and lattice induction

Michael Bj\"orklund

Stealthy point processes and lattice induction

T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Stealthy point processes are shown to exist with continuous frequency spectrum in every Euclidean dimension, and a one-dimensional converse pins down exactly which actions admit them.

desk verdict Genuinely new construction paper: first exact non-pure-point stealthy point processes, built from lattice induction, plus a clean one-dimensional converse; deserves a serious referee. read the letter →

arxiv 2607.25616 v1 pith:RVGJEBFX submitted 2026-07-28 math.PR math.DS

Michael Björklund This is my paper

classification math.PRmath.DS MSC 60G5537A1537A3043A2560G57

keywords stealthypointprocesseshyperuniformityBartlettspectrumlatticeinductionDelonecross-sectionband-limitedinterpolationspectralgapprocessrigidity

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 establishes that stealthy point processes—translation-invariant point processes whose Bartlett spectrum vanishes near the origin—are far more abundant than previously known. Its main realization theorem says that every probability-preserving R^d-action induced from a full-rank lattice admits a generating Delone cross-section whose return-time process is stealthy. In dimension one, the converse holds: an ergodic translation-invariant point process on R is lattice-induced if and only if it admits a generating stealthy Delone cross-section, and a simple integral condition on the Bartlett spectrum already forces lattice induction. As a consequence, the paper produces the first rigorously constructed translation-invariant stealthy point processes with non-pure-point Bartlett spectrum, including absolutely continuous and singular-continuous components. It also derives a sharp density bound forcing any stealthy uniformly separated process to have intensity at least a certain function of the spectral gap.

What carries the argument

The construction hinges on a band-limited entire interpolation function ψ satisfying ψ(n) = δ_{n0}, Fourier support in [−τ, τ] with τ < 1, and exponential decay estimates of order e^{2πσ|y|}(1+|x|)^{-N}. This ψ is used to encode an orbit z ↦ (T^n z) into the entire function H_z(w) = Σ b(T^n z)ψ(w−n). The paper then perturbs the 1-periodic entire function S(w) = sin(4πw) + sin(2πw) by εH_z, so that F_z = S + εH_z has uniformly Delone real zero sets P_z. The two frequencies in S serve distinct roles: the frequency 4 determines density and growth, while the frequency 2 acts as a marker preventing nonintegral translations from preserving the zero set. Band limitation gives the Fourier identity b

What would settle it

Construct an ergodic translation-invariant point process on R with positive intensity and local second moments such that ∫_{0<|ξ|<1} ξ^{-2} dσ_η(ξ) < ∞ but the translation action has no nonzero eigenvalue. Theorem 4.1 predicts that no such process exists; exhibiting one (or proving that the integral forces a nonzero eigenvalue under weaker assumptions) would directly test the converse. A more targeted falsifier: compute the Bartlett spectrum of the constructed zero-set process for a nontrivial weakly mixing Bernoulli base; if the base component σ^Z_Y vanishes identically for every choice of in

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.1, is twofold. First, for every d ≥ 1 and every full-rank lattice Γ ≤ R^d, any probability-preserving Borel R^d-space induced from a Γ-action admits a generating Delone cross-section Y whose return-time process has Bartlett spectrum vanishing on a neighborhood of the origin. Second, in dimension one, an ergodic translation-invariant point process with positive intensity and local second moments whose Bartlett spectrum satisfies ∫_{0<|ξ|<1} ξ^{-2} dσ_η(ξ) < ∞ must have a nonzero Koopman eigenvalue and therefore be lattice-induced. Together these give a complete one-dimensional characterization: an ergodic probability-preserving Borel R-space is lattice-induced if

Load-bearing premise

The entire construction rests on the existence of a real entire function ψ that equals 1 at 0, vanishes on every other integer, decays like e^{2πσ|y|}(1+|x|)^{-N}, and has Fourier support inside [−τ, τ] with τ < 1; if no such function existed, the coded perturbation could not be kept away from the marker frequencies 1 and 2, and orbit recovery from the zero set would fail.

Editorial extensions

If this is right

For every d ≥ 1, there exist ergodic translation-invariant stealthy point processes on R^d, supported on uniformly Delone configurations, whose Bartlett spectra have a nonzero absolutely continuous component and others with a nonzero singular-continuous component.
In dimension one, every ergodic stealthy translation-invariant point process with positive intensity and local second moments is lattice-induced; in particular, its translation action is never weakly mixing and never mixing.
Any uniformly separated translation-invariant point process with a spectral gap on an open neighborhood Ω of the origin must have intensity at least D(Ω), and this bound is sharp for lattice processes.
Every essentially free probability-preserving action of the p-adic group Q_p admits a generating stealthy cross-section, so mixing p-adic actions have mixing stealthy point-process realizations.
The construction gives a measurable isomorphism between a lattice-induced action and the translation action on its return-time process, so the full measure-theoretic dynamics is faithfully encoded by a stealthy Delone point process.

Reading between the lines

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

Editorial inference: The band-limited coding scheme is a general template: any action whose orbits can be injectively coded into the values of a band-limited entire function at integer points should admit a stealthy realization, potentially for other locally compact abelian groups beyond R^d and Q_p.
Editorial inference: The paper's density bound, combined with the explicit intensity 4/covol(Γ) of the constructed cross-sections, suggests the construction is not optimal; one could try to lower the intensity relative to the spectral gap by choosing a different periodic marker function or by a multi-frequency optimization.
Editorial inference: If the author's open 'cloaking' question has a positive answer—i.e., if the torus eigenvalues can be made invisible to all centered linear statistics—then stealthy point processes would exist whose Bartlett spectrum is purely continuous, which would sharply separate stealth from the torus factor structure.
Editorial inference: The one-dimensional converse gives a spectral criterion for lattice induction; the sine-kernel example shows quadratic decay of σ_η([−ε, ε]) is not enough, so identifying the exact threshold between the inverse-square condition and lattice induction would settle the boundary of the phenomenon.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I've read the paper through, and the reader's report and stress-test are fair. This is a real contribution, not a repackaging. The main theorem — every lattice-induced R^d-action admits a generating stealthy Delone cross-section — is new, and the one-dimensional converse forcing lattice induction from an inverse-square spectral condition is a sharp and satisfying statement. The construction of continuous Bartlett components (absolutely continuous from Bernoulli bases, singular-continuous from Gaussian bases) answers the open question in [LR25] in a strong sense, with the caveat they state honestly: the resulting actions are not weakly mixing, so weak mixing stealthy processes remain open in d≥2.

The analytical core is the coding by band-limited perturbations of S(w)=sin(4πw)+sin(2πw). The two-frequency trick is genuinely clever: one frequency sets density, the other kills small nonintegral translations, and the Fourier-support separation τ<1 makes orbit recovery work. The stress-test note is right that Lemma 5.2 is load-bearing. I checked the interpolation construction via θ=χ/D and the partition identity; it works, and the decay estimate follows from compact support plus integration by parts with σ>τ absorbing polynomial factors. The Rouché/Hadamard/contour residue chain in Sections 5–7 is intricate but coherent. The one-dimensional converse via the L²-coboundary criterion and exponentiation of an integer-valued coboundary is elegant and, as far as I can tell, sound.

Soft spots are minor. The proofs lean heavily on external technical results (BSW18, Landau's density theorem, Kechris's lacunary sections); each is cited clearly, but the paper would be even stronger with a footnote spelling out the exact normalization in Landau (they do address it). Some steps in Section 7 — the L²-differentiability of the linear statistics in s and the implicit-function arguments — are sketched rather than written out in full. The reader's MODERATE confidence is appropriate: the estimates are plausible and nothing looks broken, but a full formal check of the contour and decay estimates would be needed to rule out subtle gaps. The paper's own remarks about open cloaking and the d≥2 weak-mixing question are honest and frame the limits correctly.

Who should read this: ergodic theorists, point-process people, and anyone working on hyperuniformity or Fourier quasicrystals. It answers a posed open problem and gives new tools (the band-limited coding scheme) likely to be reused. The AI-use statement and self-citations don't undermine anything. I'd send it to a serious referee, and I'd bring it to a reading group myself.

Referee Report

0 major / 4 minor

Summary. The paper proves realization and rigidity theorems for stealthy point processes. The main result (Theorem 1.1) states that every lattice-induced probability-preserving R^d-action has a generating stealthy Delone cross-section, and, conversely, in dimension one an ergodic translation-invariant point process with positive intensity and local second moments whose Bartlett spectrum satisfies the inverse-square integrability condition near the origin is lattice-induced. The construction encodes the orbit of a Borel automorphism T in a band-limited entire function H_z, perturbs the periodic sine function S(w)=sin(4πw)+sin(2πw), and takes the real zero set; the two frequencies in S give Delone geometry, a Fourier gap on (-1,1), and orbit recovery via Hadamard factorization and Fourier coefficient comparison at frequencies 1 and 2. Layering this one-dimensional coding over Z^{d-1} and applying a linear change of coordinates gives the higher-dimensional and arbitrary-lattice cases. Further theorems show that the inducing spectral type can be made visible in the Bartlett spectrum (continuous components, including absolutely continuous and singular-continuous), prove a density lower bound from a spectral gap, and construct stealthy positive random measures and Q_p cross-sections.

Significance. If the results hold, they provide the first rigorous translation-invariant stealthy point processes on R^d with non-pure-point Bartlett spectrum, strongly resolving a question that had been open for d≥2. The proof is genuinely constructive and uses standard, well-matched tools: Rouché's theorem, Hadamard factorization, contour residue arguments, the L^2-coboundary criterion, Landau's sampling theorem, and Kechris's lacunary-section theorem. I found no circularity: the small parameter ε is not fitted to any target, and Theorem 1.1(2) derives lattice induction from a structural integrability condition on the Bartlett spectrum. The stress-test concern about Lemma 5.2 is appropriately addressed: τ<1 is explicitly obtained in the construction, and the spectral separation in Lemma 5.5 and Proposition 5.7 indeed rests on it. The paper also gives a clean one-dimensional converse, a sharp density bound with a matching lattice example, and interesting contrasts with the Z-rigidity theorem of Borichev–Sodin–Weiss.

minor comments (4)

[§5.3, Lemma 5.5] The displayed Fourier-support formula is terse. It would help to state explicitly that {1,3,4} are the frequencies contributed by e^{2πiw}, -e^{6πiw}, -e^{8πiw}, and that the interval [2-τ,2+τ] comes from the shift e^{4πiw}H_z(w). This is the key separation step, and making the frequency bookkeeping explicit would improve readability.
[§6.1, Proposition 6.2] In the equivariance calculation, the notation T^k \widehat{m+\ell}.z is ambiguous. Define e_{m+\ell} explicitly and write T^k(e_{m+\ell}.z) instead.
[§10.1, Theorem 10.1(1)] The sentence 'Since G/K is discrete, Y_x is locally finite, and N^Y_f is uniformly bounded for each fixed f∈C_c(G)' appears twice in the proof. The duplicate sentence can be removed.
[§4.1, Theorem 4.1] In the display dς_{A_a}(ξ)=|\widehat{1_{[0,a)}}(ξ)|^2 dση(ξ), it would be useful to note explicitly that ση({0})=0, so the value of \widehat{1_{[0,a)}} at ξ=0 is immaterial. This is implicit in the proof but worth stating.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The derivation chain is self-contained. The main construction starts from a lattice-induced action and an arbitrary Borel base (Z,ν); the free parameters (ε, σ, τ, s) are structural constants fixed by compactness, Rouché estimates, and interpolation arguments, and are never fitted to the target stealthiness or spectral conclusions. Lemma 5.2 is proved in-line from θ=χ/D and the partition identity Σ_k θ(ξ+k)=1; the crucial support bound τ<1 follows from the prescribed range 1/2<σ<1 and is not imported from the theorem being proved. The Fourier identity bδPz=4δ0 on (−1,1) is derived by residue calculus and Fourier-support separation (Lemma 5.5), and orbit recovery follows from Hadamard factorization and comparison at frequencies 1 and 2; none of these steps assumes the conclusion. The one-dimensional converse (Theorem 4.1) uses the standard L²-coboundary criterion and the integer-valuedness of interval counts to construct an eigenfunction, then applies Lemma 3.1, which is proved in the paper; the eigenvalue is derived, not assumed. No prediction is statistically forced by a fitted parameter, and no load-bearing uniqueness or existence claim rests on an unverified self-citation. The self-citations that occur ([BHK25], [ABC25], [BB25], [BB26], [BH24]) serve as background formalism or as comparison examples and do not carry the central argument. Open questions are stated as such and do not conceal a circular step.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

No parameters are fitted to empirical or target data; the listed ε, s, and a are existential smallness or phase-choice parameters. The proofs rely on standard tools from complex analysis, descriptive set theory, ergodic theory, and sampling theory. No new entities such as particles, forces, or dimensions are postulated.

free parameters (3)

epsilon
Sufficiently small perturbation amplitude in F_z = S + εH_z; chosen in Proposition 5.3 so that Rouché's theorem controls the zero sets. Existential, not fitted to data.
s
Nonzero small parameter in Section 7, retained to make a base spectral component visible in the Bartlett spectrum. Chosen after Lemma 7.6; no fitted value.
a
Real number chosen in the proof of Theorem 4.1 with ρa∉Z, so that the interval count produces a non-trivial eigenfunction. Existence is guaranteed for any ergodic process; not a fitted constant.

assumptions (8)

standard math Standard descriptive set theory facts: Lusin–Novikov theorem, Lusin–Souslin theorem, Borel transversals for countable-section relations.
Used throughout Sections 2–3 and 10 to make return-time maps Borel, cross-sections measurable, and inverse maps Borel.
standard math Rouché's theorem, Hadamard factorization for entire functions of order at most one, and the Paley–Wiener theorem for band-limited functions.
Central to Sections 5–6: zeros are controlled via Rouché, orbit recovery uses Hadamard factorization, and band limitation gives the Fourier identities.
standard math L^2-coboundary criterion for unitary operators: F∈Ran(U−I) iff ∫|z−1|^{-2}dν_F(z)<∞.
Invoked in Theorem 4.1 to pass from the inverse-square Bartlett condition to an eigenfunction of the time-a map.
standard math Landau's necessary density theorem for stable sampling sets in Paley–Wiener spaces.
Used in Section 8 to convert exact sampling into the intensity lower bound D(Ω).
standard math Wiener pointwise ergodic theorem for R^d-actions.
Used in Theorem 8.1 to identify the Beurling density of almost every configuration with the intensity ρ.
standard math Kechris's complete lacunary Borel section theorem for free Borel actions of locally compact second-countable groups.
Used in Theorem 10.1 to produce stealthy cross-sections over compact open subgroups.
domain assumption Probability-preserving Borel actions on standard Borel spaces can be restricted to invariant conull Borel subsets on which the maps in question are well defined and the action is free where needed.
This underlies the entire cross-section formalism; the Q_p construction explicitly restricts to a free action before applying Kechris's theorem.
domain assumption In Theorems 1.2 and 7.4, the base Γ-action is assumed nontrivial and weakly mixing.
A stated hypothesis of the theorem, needed to ensure the lifted maximal spectral type is atomless and the base component is continuous.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Stealthy point processes and lattice induction." pith.science (2026). https://pith.science/paper/RVGJEBFX

@misc{pith2026260725616,
  author       = {Pith},
  title        = {Pith review of: Stealthy point processes and lattice induction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RVGJEBFX}},
  note         = {Machine review of arXiv:2607.25616}
}

abstract

We prove a realization theorem for stealthy point processes based on lattice induction, together with a converse in dimension one. Every probability-preserving $\mathbb R^d$-action induced from a full-rank lattice admits a generating Delone cross-section whose Bartlett spectrum vanishes on a neighborhood of the origin. The construction can be chosen so that first-order linear statistics detect a nonzero part of the inducing spectral type. Bernoulli bases yield a nonzero absolutely continuous component, while weakly mixing bases of singular maximal spectral type yield a nonzero singular-continuous component. To our knowledge, these are the first rigorously constructed translation-invariant stealthy point processes on $\mathbb R^d$ with non-pure-point Bartlett spectrum. Conversely, let $\eta$ be an ergodic translation-invariant point process on $\mathbb R$ with positive intensity and local second moments. If \[ \int_{0<|\xi|<1}\frac{1}{\xi^2}\,d\sigma_\eta(\xi)<\infty, \] then its translation action has a nonzero eigenvalue and is lattice-induced. Consequently, an ergodic probability-preserving Borel $\mathbb R$-space is lattice-induced if and only if it admits a generating stealthy Delone cross-section. This should be compared with a theorem of Borichev, Sodin and Weiss stating that a translation-invariant point process on $\mathbb Z$ with proper spectral support is periodic.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

17 extracted references · 3 linked inside Pith

[1]

[AKKV25] Alon, L.; Kummer, M.; Kurasov, P.; Vinzant, C.Higher dimensional Fourier quasicrystals from Lee–Yang varieties. Invent. Math. 239 (2025), no. 1, 321–376. [ABC25] Avraham-Re’em, N.; Björklund, M.; Cullman, R. Locally integrable cross sections and their intersection covolume. arXiv:2509.20836v3 [math.DS],

arXiv 2025
[10]

Maximal rigidity of random measures and uniqueness pairs: stealthy processes, quasicrystals and periodicity

[LR26] Lachièze-Rey, R. Maximal rigidity of random measures and uniqueness pairs: stealthy processes, quasicrystals and periodicity. J. Stat. Phys. 193 (2026), article no

2026
[13]

F.Crystalline measures and mean-periodic functions.Trans

[Mey21] Meyer, Y. F.Crystalline measures and mean-periodic functions.Trans. R. Norw. Soc. Sci. Lett. 2021 (2021), no. 2, 5–30. STEALTHY POINT PROCESSES AND LATTICE INDUCTION 57 [MST24] Morse, P. K.; Steinhardt, P. J.; Torquato, S.Ordered and disordered stealthy hyperuniform point patterns across spatial dimensions. Phys. Rev. Research 6 (2024), 033260. [O...

2021
[15]

Regularization of a stationary point process by a stationary increments perturbation

[TLS26] Thomassey, L.; Lachièze-Rey, R.; Shapira, A. Regularization of a stationary point process by a stationary increments perturbation. arXiv:2602.19773 [math.PR],

arXiv
[16]

H.Local density fluctuations, hyperuniformity, and order metrics

[TS03] Torquato, S.; Stillinger, F. H.Local density fluctuations, hyperuniformity, and order metrics. Phys. Rev. E 68 (2003), 041113. [Tor18] Torquato, S.Hyperuniform states of matter. Phys. Rep. 745 (2018), 1–95. [UST04] Uche, O. U.; Stillinger, F. H.; Torquato, S.Constraints on collective density variables: two dimensions. Phys. Rev. E 70 (2004), 046122...

2003
[46]

BJÖRKLUND [BH24] Björklund, M.; Hartnick, T.Hyperuniformity and non-hyperuniformity of quasicrystals

56 M. BJÖRKLUND [BH24] Björklund, M.; Hartnick, T.Hyperuniformity and non-hyperuniformity of quasicrystals. Math. Ann. 389 (2024), no. 1, 365–426. [BHK25] Björklund, M.; Hartnick, T.; Karasik, Y.Intersection spaces and multiple transverse recurrence.J. Anal. Math. 156 (2025), no. 1, 97–150. [BSW18] Borichev, A.; Sodin, M.; Weiss, B.Spectra of stationary p...

2024
[88]

J.Necessary density conditions for sampling and interpolation of certain entire functions.Acta Math

[Lan67] Landau, H. J.Necessary density conditions for sampling and interpolation of certain entire functions.Acta Math. 117 (1967), 37–52. [Len73] Lenard, A.Correlation functions and the uniqueness of the state in classical statistical mechanics. Comm. Math. Phys. 30 (1973), 35–44. [Len75a] Lenard, A.States of classical statistical mechanical systems of i...

1967
[1990]

P.The Fourier transform of the characteristic function of a set, vanishing on an interval

[Kar83] Kargaev, P. P.The Fourier transform of the characteristic function of a set, vanishing on an interval. Math. USSR-Sb. 45 (1983), no. 3, 397–410. [KT06] Katok, A.; Thouvenot, J.-P.Spectral properties and combinatorial constructions in ergodic theory. InHandbook of Dynamical Systems, vol. 1B, Elsevier, Amsterdam, 2006, 649–743. [Kec92] Kechris, A. S...

1983

Show all 17 references

[1995]

N.; Lev, N.On non-periodic tilings of the real line by a function

[KL16] Kolountzakis, M. N.; Lev, N.On non-periodic tilings of the real line by a function. Int. Math. Res. Not. IMRN 2016, no. 15, 4588–4601. [KP02] Krantz, S. G.; Parks, H. R.A Primer of Real Analytic Functions. Second edition, Birkhäuser Advanced Texts: Basler Lehrbücher, Bi...

2016
[1996]

[LK26] Lotz, L.; Klatt, M. A. Persistence of asymptotic variance under transport: from hyperfluctuation to stealthy hyperuniformity. arXiv:2605.22803 [math.PR],

arXiv
[1997]

[CL21] Cohen, G.; Lin, M.Joint and double coboundaries of commuting contractions.Indiana Univ. Math. J. 70 (2021), no. 4, 1355–1394. [CL04] Costin, O.; Lebowitz, J. L.On the construction of particle distributions with specified single and pair densities. J. Phys. Chem. B 108 (...

2021
[2001]

H.; Torquato, S.Ground states of stealthy hyperuniform potentials: I

[ZST15a] Zhang, G.; Stillinger, F. H.; Torquato, S.Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations. Phys. Rev. E 92 (2015), 022119. [ZST15b] Zhang, G.; Stillinger, F. H.; Torquato, S.Ground states of stealthy hyperuniform potentials: I...

2015
[2002]

L.; Speer, E

[KLS07] Kuna, T.; Lebowitz, J. L.; Speer, E. R.Realizability of point processes. J. Stat. Phys. 129 (2007), 417–439. [KLS11] Kuna, T.; Lebowitz, J. L.; Speer, E. R.Necessary and sufficient conditions for realizability of point processes. Ann. Appl. Probab. 21 (2011), 1253–1281...

2007 arXiv
[2003]

[DM08] Deng, X.; Moody, R. V. Dworkin’s argument revisited: point processes, dynamics, diffraction, and correlations. J. Geom. Phys. 58 (2008), no. 4, 506–541. [FW89] Furstenberg, H.; Weiss, B.On almost1-1extensions. Israel J. Math. 65 (1989), no. 3, 311–322. [GL18] Ghosh, S.;...

2008
[2016]

Ergodic Theory Dynam

[Osa21] Osada, S.Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes. Ergodic Theory Dynam. Systems 41 (2021), no. 12, 3807–3820. [SS03] Stein, E. M.; Shakarchi, R.Complex Analysis. Princeton Lectures in Analysis, v...

2021
[2025]

Ann.394(2026), Article

[BB26] Björklund, M.; Byléhn, M.;Hyperuniformity of random measures on Euclidean and hyperbolic spaces, Math. Ann.394(2026), Article

2026
[2026]

Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra

[BL04] Baake, M.; Lenz, D. Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra. Ergodic Theory Dynam. Systems 24 (2004), no. 6, 1867–1893. [BLvE15] Baake, M.; Lenz, D.; van Enter, A. C. D. Dynamical versus diffraction spectrum for st...

2004 arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

[AKKV25] Alon, L.; Kummer, M.; Kurasov, P.; Vinzant, C.Higher dimensional Fourier quasicrystals from Lee–Yang varieties. Invent. Math. 239 (2025), no. 1, 321–376. [ABC25] Avraham-Re’em, N.; Björklund, M.; Cullman, R. Locally integrable cross sections and their intersection covolume. arXiv:2509.20836v3 [math.DS],

arXiv 2025

[2] [10]

Maximal rigidity of random measures and uniqueness pairs: stealthy processes, quasicrystals and periodicity

[LR26] Lachièze-Rey, R. Maximal rigidity of random measures and uniqueness pairs: stealthy processes, quasicrystals and periodicity. J. Stat. Phys. 193 (2026), article no

2026

[3] [13]

F.Crystalline measures and mean-periodic functions.Trans

[Mey21] Meyer, Y. F.Crystalline measures and mean-periodic functions.Trans. R. Norw. Soc. Sci. Lett. 2021 (2021), no. 2, 5–30. STEALTHY POINT PROCESSES AND LATTICE INDUCTION 57 [MST24] Morse, P. K.; Steinhardt, P. J.; Torquato, S.Ordered and disordered stealthy hyperuniform point patterns across spatial dimensions. Phys. Rev. Research 6 (2024), 033260. [O...

2021

[4] [15]

Regularization of a stationary point process by a stationary increments perturbation

[TLS26] Thomassey, L.; Lachièze-Rey, R.; Shapira, A. Regularization of a stationary point process by a stationary increments perturbation. arXiv:2602.19773 [math.PR],

arXiv

[5] [16]

H.Local density fluctuations, hyperuniformity, and order metrics

[TS03] Torquato, S.; Stillinger, F. H.Local density fluctuations, hyperuniformity, and order metrics. Phys. Rev. E 68 (2003), 041113. [Tor18] Torquato, S.Hyperuniform states of matter. Phys. Rep. 745 (2018), 1–95. [UST04] Uche, O. U.; Stillinger, F. H.; Torquato, S.Constraints on collective density variables: two dimensions. Phys. Rev. E 70 (2004), 046122...

2003

[6] [46]

BJÖRKLUND [BH24] Björklund, M.; Hartnick, T.Hyperuniformity and non-hyperuniformity of quasicrystals

56 M. BJÖRKLUND [BH24] Björklund, M.; Hartnick, T.Hyperuniformity and non-hyperuniformity of quasicrystals. Math. Ann. 389 (2024), no. 1, 365–426. [BHK25] Björklund, M.; Hartnick, T.; Karasik, Y.Intersection spaces and multiple transverse recurrence.J. Anal. Math. 156 (2025), no. 1, 97–150. [BSW18] Borichev, A.; Sodin, M.; Weiss, B.Spectra of stationary p...

2024

[7] [88]

J.Necessary density conditions for sampling and interpolation of certain entire functions.Acta Math

[Lan67] Landau, H. J.Necessary density conditions for sampling and interpolation of certain entire functions.Acta Math. 117 (1967), 37–52. [Len73] Lenard, A.Correlation functions and the uniqueness of the state in classical statistical mechanics. Comm. Math. Phys. 30 (1973), 35–44. [Len75a] Lenard, A.States of classical statistical mechanical systems of i...

1967

[8] [1990]

P.The Fourier transform of the characteristic function of a set, vanishing on an interval

[Kar83] Kargaev, P. P.The Fourier transform of the characteristic function of a set, vanishing on an interval. Math. USSR-Sb. 45 (1983), no. 3, 397–410. [KT06] Katok, A.; Thouvenot, J.-P.Spectral properties and combinatorial constructions in ergodic theory. InHandbook of Dynamical Systems, vol. 1B, Elsevier, Amsterdam, 2006, 649–743. [Kec92] Kechris, A. S...

1983