HEXAGONbeta

COUNTABLY INFINITE-DIMENSIONAL SPACES OF CRYSTALLINE MEASURES

Primarily AI-generated textHuman understanding: no partsmath.FA — Functional Analysismath.CA — Classical Analysis and ODEs

Contributed by Jamie Martin ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

We construct a symmetric locally finite set Λ ⊂ R for which the space MΛ of crystalline measures with support and spectrum in Λ has countably infinite Hamel dimension; Meyer asked what the dimension of such spaces can be. More precisely, there is a sequence of crystalline measures such that every element of MΛ is a unique finite linear combination of them. Every layer of MΛ of fixed Hermite–Sobolev order is finite-dimensional, but the layer dimensions are unbounded; in one construction they grow linearly, and in general their growth can be prescribed up to a linear change of the order. (For every locally finite set the dimension is finite, countably infinite or equal to the continuum, and the countable case is exactly this behaviour of the layers.) The construction uses clusters of atoms that imitate derivatives of Dirac masses, which produce one-dimensional blocks of prescribed Hermite order. The main difficulty is to show that the union of infinitely many blocks supports no other crystalline measures. We prove this by placing the blocks at random scales, separated by rapidly growing gaps, and showing that every element of fixed Hermite order splits into finitely many block components. We also obtain disjoint sets A, B such that the crystalline measures with support in B and spectrum in A form a countably infinite-dimensional space, while MA = MB = {0}. The existence statement, without the symmetry of Λ, has been formalised and machine-checked in Lean 4; the formal development is recorded in Palomar

Provenance statement

Large efforts have gone in to ensure the work is correct, verified and self-consistent, without having to understand all of the details. This has been handled by using verification tooling, verification calculations and consistency checks using multiple AI models.

Formalizations

Tools used

OpenAI
CodexVersion Sol 5.6
OpenAI
CodexVersion Astra 6
OpenAI
ChatGPTVersion Sol 5.6 Pro

References

  1. A. Adve, Density criteria for Fourier uniqueness phenomena in Rd\R^d , preprint, arXiv:2306.07475 (https://arxiv.org/abs/2306.07475) (2023).arXiv
  2. C. Alfes, P. Kiefer and J. Mazáč, Measures, modular forms, and summation formulas of Poisson type , Comm. Math. Phys. 406 (2025), no. 6, Paper No. 137. doi:10.1007/s00220-025-05313-6 (https://doi.org/10.1007/s00220-025-05313-6).DOI
  3. L. Alon, A. Cohen and C. Vinzant, Every real-rooted exponential polynomial is the restriction of a Lee--Yang polynomial , J. Funct. Anal. 286 (2024), no. 2, Paper No. 110226. doi:10.1016/j.jfa.2023.110226 (https://doi.org/10.1016/j.jfa.2023.110226).DOI
  4. L. Alon, M. Kummer, P. Kurasov and C. Vinzant, Higher dimensional Fourier quasicrystals from Lee--Yang varieties , Invent. Math. 239 (2025), no. 1, 321--376. doi:10.1007/s00222-024-01307-8 (https://doi.org/10.1007/s00222-024-01307-8).DOI
  5. L. Alon and C. Vinzant, Gap distributions of Fourier quasicrystals with integer weights via Lee--Yang polynomials , Rev. Mat. Iberoam. 40 (2024), no. 6, 2203--2250. doi:10.4171/RMI/1485 (https://doi.org/10.4171/RMI/1485).DOI
  6. M. Baake and U. Grimm, Aperiodic Order. Vol. 1: A Mathematical Invitation , Encyclopedia Math. Appl. 149, Cambridge Univ. Press, Cambridge, 2013. doi:10.1017/CBO9781139025256 (https://doi.org/10.1017/CBO9781139025256).DOI
  7. M. Baake, T. Spindeler and N. Strungaru, On eigenmeasures under Fourier transform , J. Fourier Anal. Appl. 29 (2023), Paper No. 65. doi:10.1007/s00041-023-10045-z (https://doi.org/10.1007/s00041-023-10045-z).DOI
  8. M. Baake and N. Strungaru, A note on tempered measures , Colloq. Math. 172 (2023), 15--30. doi:10.4064/cm8796-7-2022 (https://doi.org/10.4064/cm8796-7-2022).DOI
  9. D. Berghaus, A. Bondarenko, D. Radchenko, K. Seip and Q. Sun, The basis functions of Fourier interpolation , preprint, arXiv:2512.18677 (https://arxiv.org/abs/2512.18677) (2025).arXiv
  10. A. Bondarenko, D. Radchenko and K. Seip, Fourier interpolation with zeros of zeta and LL-functions , Constr. Approx. 57 (2023), 405--461. doi:10.1007/s00365-022-09599-w (https://doi.org/10.1007/s00365-022-09599-w).DOI
Show all 72 referencesShow fewer references
  1. A. Bondarenko and K. Seip, Fourier-invariant functions with dense zero sets , preprint, arXiv:2608.13468 (https://arxiv.org/abs/2608.13468) (2026).arXiv
  2. P. Boyvalenkov and S. Favorov, Growth of masses of crystalline measures , Studia Math. 290 (2026), 109--119. doi:10.4064/sm250325-8-1 (https://doi.org/10.4064/sm250325-8-1).DOI
  3. A. Carbery, M. Christ and J. Wright, Multidimensional van der Corput and sublevel set estimates , J. Amer. Math. Soc. 12 (1999), 981--1015. doi:10.1090/S0894-0347-99-00309-4 (https://doi.org/10.1090/S0894-0347-99-00309-4).DOI
  4. H. Cohn, From sphere packing to Fourier interpolation , Bull. Amer. Math. Soc. (N.S.) 61 (2024), no. 1, 3--22. doi:10.1090/bull/1813 (https://doi.org/10.1090/bull/1813).DOI
  5. A. Córdoba, Dirac combs , Lett. Math. Phys. 17 (1989), 191--196. doi:10.1007/BF00401584 (https://doi.org/10.1007/BF00401584).DOI
  6. F. Dyson, Birds and frogs , Notices Amer. Math. Soc. 56 (2009), no. 2, 212--223.
  7. M. Eidelheit, Zur Theorie der Systeme linearer Gleichungen , Studia Math. 6 (1936), 139--148. doi:10.4064/sm-6-1-139-148 (https://doi.org/10.4064/sm-6-1-139-148).DOI
  8. S. Yu. Favorov, Fourier quasicrystals and Lagarias' conjecture , Proc. Amer. Math. Soc. 144 (2016), 3527--3536. doi:10.1090/proc/12979 (https://doi.org/10.1090/proc/12979).DOI
  9. S. Yu. Favorov, Uniqueness theorems for Fourier quasicrystals and temperate distributions with discrete support , Proc. Amer. Math. Soc. 149 (2021), 4431--4440. doi:10.1090/proc/15546 (https://doi.org/10.1090/proc/15546).DOI
  10. S. Yu. Favorov, Temperate distributions with locally finite support and spectrum on Euclidean spaces , Israel J. Math. 255 (2023), 705--728. doi:10.1007/s11856-022-2450-z (https://doi.org/10.1007/s11856-022-2450-z).DOI
  11. S. Yu. Favorov, Almost periodic distributions and crystalline measures , Mat. Stud. 61 (2024), no. 1, 97--108. doi:10.30970/ms.61.1.97-108 (https://doi.org/10.30970/ms.61.1.97-108).DOI
  12. S. Yu. Favorov, A crystalline measure that is not a Fourier quasicrystal , Anal. Math. 50 (2024), no. 2, 455--462. doi:10.1007/s10476-024-00031-y (https://doi.org/10.1007/s10476-024-00031-y).DOI
  13. F. Gonçalves, A classification of Fourier summation formulas and crystalline measures , preprint, arXiv:2312.11185 (https://arxiv.org/abs/2312.11185) (2023; v3, 2026).arXiv
  14. F. Gonçalves and G. Vedana, A complete classification of Fourier summation formulas on the real line , preprint, arXiv:2504.02741 (https://arxiv.org/abs/2504.02741) (2025).arXiv
  15. A. P. Guinand, Concordance and the harmonic analysis of sequences , Acta Math. 101 (1959), 235--271. doi:10.1007/BF02559556 (https://doi.org/10.1007/BF02559556).DOI
  16. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers , 6th ed., revised by D. R. Heath-Brown and J. H. Silverman, Oxford University Press, Oxford, 2008.
  17. R. Hartshorne, Algebraic Geometry , Grad. Texts in Math. 52 , Springer, New York, 1977. doi:10.1007/978-1-4757-3849-0 (https://doi.org/10.1007/978-1-4757-3849-0).DOI
  18. L. Hörmander, The Analysis of Linear Partial Differential Operators I , reprint of the second (1990) edition, Classics in Mathematics, Springer, Berlin, 2003. doi:10.1007/978-3-642-61497-2 (https://doi.org/10.1007/978-3-642-61497-2).DOI
  19. J.-P. Kahane and S. Mandelbrojt, Sur l'équation fonctionnelle de Riemann et la formule sommatoire de Poisson , Ann. Sci. École Norm. Sup. (3) 75 (1958), 57--80. doi:10.24033/asens.1066 (https://doi.org/10.24033/asens.1066).DOI
  20. M. N. Kolountzakis, Fourier pairs of discrete support with little structure , J. Fourier Anal. Appl. 22 (2016), no. 1, 1--5. doi:10.1007/s00041-015-9416-z (https://doi.org/10.1007/s00041-015-9416-z).DOI
  21. A. Kulikov, Fourier interpolation and time-frequency localization , J. Fourier Anal. Appl. 27 (2021), Paper No. 58. doi:10.1007/s00041-021-09861-y (https://doi.org/10.1007/s00041-021-09861-y).DOI
  22. A. Kulikov, F. Nazarov and M. Sodin, Fourier uniqueness and non-uniqueness pairs , J. Math. Phys. Anal. Geom. 21 (2025), no. 1, 84--130. doi:10.15407/mag21.01.04 (https://doi.org/10.15407/mag21.01.04).DOI
  23. P. Kurasov and P. Sarnak, Stable polynomials and crystalline measures , J. Math. Phys. 61 (2020), no. 8, 083501. doi:10.1063/5.0012286 (https://doi.org/10.1063/5.0012286).DOI
  24. H. E. Lacey, The Hamel dimension of any infinite dimensional separable Banach space is cc , Amer. Math. Monthly 80 (1973), 298. doi:10.2307/2318458 (https://doi.org/10.2307/2318458).DOI
  25. J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction , in: Directions in Mathematical Quasicrystals (M. Baake and R. V. Moody, eds.), CRM Monogr. Ser. 13, Amer. Math. Soc., Providence, RI, 2000, 61--93. doi:10.1090/crmm/013/03 (https://doi.org/10.1090/crmm/013/03).DOI
  26. N. Lev and A. Olevskii, Measures with uniformly discrete support and spectrum , C. R. Math. Acad. Sci. Paris 351 (2013), 599--603. doi:10.1016/j.crma.2013.09.007 (https://doi.org/10.1016/j.crma.2013.09.007).DOI
  27. N. Lev and A. Olevskii, Quasicrystals and Poisson's summation formula , Invent. Math. 200 (2015), no. 2, 585--606. doi:10.1007/s00222-014-0542-z (https://doi.org/10.1007/s00222-014-0542-z).DOI
  28. N. Lev and A. Olevskii, Quasicrystals with discrete support and spectrum , Rev. Mat. Iberoam. 32 (2016), no. 4, 1341--1352. doi:10.4171/RMI/920 (https://doi.org/10.4171/RMI/920).DOI
  29. N. Lev and A. Olevskii, Fourier quasicrystals and discreteness of the diffraction spectrum , Adv. Math. 315 (2017), 1--26. doi:10.1016/j.aim.2017.05.015 (https://doi.org/10.1016/j.aim.2017.05.015).DOI
  30. N. Lev and G. Reti, Poisson summation formulas involving the sum-of-squares function , Israel J. Math. 246 (2021), 403--421. doi:10.1007/s11856-021-2252-8 (https://doi.org/10.1007/s11856-021-2252-8).DOI
  31. N. Lev and G. Reti, Crystalline temperate distributions with uniformly discrete support and spectrum , J. Funct. Anal. 281 (2021), no. 4, Paper No. 109072. doi:10.1016/j.jfa.2021.109072 (https://doi.org/10.1016/j.jfa.2021.109072).DOI
  32. T. K. Lysen, Critical and asymmetric Fourier uniqueness pairs , preprint, arXiv:2509.17600 (https://arxiv.org/abs/2509.17600) (2025).arXiv
  33. G. W. Mackey, On infinite-dimensional linear spaces , Trans. Amer. Math. Soc. 57 (1945), 155--207. doi:10.1090/S0002-9947-1945-0012204-1 (https://doi.org/10.1090/S0002-9947-1945-0012204-1).DOI
  34. J. Martin, A strongly tempered Meyer space of countably infinite Hamel dimension , Lean 4 formalization, GitHub repository (2026), https://github.com/balancedscorpion/countably-infinite-dimension-strong-meyer-spaces, commit 4d3163a .link
  35. J. Martin and L. Shen, A complete crystalline-measure space of countably infinite Hamel dimension , Lean 4 formalization, GitHub repository (2026), https://github.com/balancedscorpion/countably-infinite-dimension-meyer-spaces, commit 0de6196 .link
  36. J. Mazáč, C. Richard and N. Strungaru, On almost periodicity in crystalline measures , preprint, arXiv:2605.23884 (https://arxiv.org/abs/2605.23884) (2026).arXiv
  37. R. Meise and D. Vogt, Einführung in die Funktionalanalysis , Vieweg, Braunschweig, 1992; English transl.: Introduction to Functional Analysis , Oxford Grad. Texts Math. 2, Clarendon Press, Oxford, 1997.
  38. Y. F. Meyer, Measures with locally finite support and spectrum , Proc. Natl. Acad. Sci. USA 113 (2016), no. 12, 3152--3158. doi:10.1073/pnas.1600685113 (https://doi.org/10.1073/pnas.1600685113).DOI
  39. Y. Meyer, Measures with locally finite support and spectrum , Rev. Mat. Iberoam. 33 (2017), no. 3, 1025--1036. doi:10.4171/RMI/962 (https://doi.org/10.4171/RMI/962).DOI
  40. Y. Meyer, Guinand's measures are almost periodic distributions , Bull. Hellenic Math. Soc. 61 (2017), 11--20.
  41. Y. F. Meyer, Crystalline measures and mean-periodic functions , Trans. R. Norw. Soc. Sci. Lett. 2021 , no. 2, 5--30.
  42. Y. Meyer, Mesures cristallines et applications [d'après P. Kurasov, N. Lev, A. Olevskii, P. Sarnak et M. Viazovska] , Séminaire Bourbaki, exposé no. 1194 (avril 2022), Astérisque 438 (2023). doi:10.24033/ast.1195 (https://doi.org/10.24033/ast.1195).DOI
  43. Y. Meyer, Multidimensional crystalline measures , Trans. R. Norw. Soc. Sci. Lett. 2023 , no. 1.
  44. Y. Meyer, Crystalline measures in two dimensions , Publ. Mat. 67 (2023), 469--480. doi:10.5565/publmat6712310 (https://doi.org/10.5565/publmat6712310).DOI
  45. Y. Meyer, Curved model sets and crystalline measures , in: Theoretical Physics, Wavelets, Analysis, Genomics, Appl. Numer. Harmon. Anal., Birkhäuser, Cham, 2023, 389--407. doi:10.1007/978-3-030-45847-8_17 (https://doi.org/10.1007/978-3-030-45847-8_17).DOI
  46. A. Olevskii and A. Ulanovskii, A simple crystalline measure , preprint, arXiv:2006.12037 (https://arxiv.org/abs/2006.12037) (2020).arXiv
  47. A. Olevskii and A. Ulanovskii, Fourier quasicrystals with unit masses , C. R. Math. Acad. Sci. Paris 358 (2020), no. 11--12, 1207--1211. doi:10.5802/crmath.142 (https://doi.org/10.5802/crmath.142).DOI
  48. Palomar record PALOMAR-2026-08-25-000001 , version 1: J. Martin and L. Shen, Lean formalization of the classification of crystalline measures on the shifted square-root sets {±n+θ:n≥0}\{\pm\sqrt{n+\theta}:n\ge0\} (theorems shiftedSquareRootClassification and shiftedSquareRootDimensionFormula ), registered 25 August 2026, https://palomar-registry.org/entry?id=PALOMAR-2026-08-25-000001&version=1.link
  49. Palomar record PALOMAR-2026-09-30-000036 , version 1, of the formal development MS26 (theorem exhaustiveCardinalClaim ), submitted 30 September 2026, https://palomar-registry.org/entry?id=PALOMAR-2026-09-30-000036&version=1.link
  50. Palomar record PALOMAR-2026-10-08-000002 , version 1, of the formal development Mar26 (theorem strongCardinalClaim ), registered 8 October 2026, https://palomar-registry.org/entry?id=PALOMAR-2026-10-08-000002&version=1.link
  51. D. Radchenko and M. Viazovska, Fourier interpolation on the real line , Publ. Math. Inst. Hautes Études Sci. 129 (2019), 51--81. doi:10.1007/s10240-018-0101-z (https://doi.org/10.1007/s10240-018-0101-z).DOI
  52. J. P. G. Ramos and M. Sousa, Fourier uniqueness pairs of powers of integers , J. Eur. Math. Soc. 24 (2022), no. 12, 4327--4351. doi:10.4171/JEMS/1194 (https://doi.org/10.4171/JEMS/1194).DOI
  53. J. P. G. Ramos and M. Sousa, Perturbed interpolation formulae and applications , Anal. PDE 16 (2023), no. 10, 2327--2384. doi:10.2140/apde.2023.16.2327 (https://doi.org/10.2140/apde.2023.16.2327).DOI
  54. J. P. G. Ramos and M. Stoller, Perturbed Fourier uniqueness and interpolation results in higher dimensions , J. Funct. Anal. 282 (2022), Paper No. 109448. doi:10.1016/j.jfa.2022.109448 (https://doi.org/10.1016/j.jfa.2022.109448).DOI
  55. M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis , 2nd ed., Academic Press, New York, 1980.
  56. C. Richard and N. Strungaru, Pure point diffraction and Poisson summation , Ann. Henri Poincaré 18 (2017), no. 12, 3903--3931. doi:10.1007/s00023-017-0620-z (https://doi.org/10.1007/s00023-017-0620-z).DOI
  57. B. Simon, Distributions and their Hermite expansions , J. Math. Phys. 12 (1971), 140--148. doi:10.1063/1.1665472 (https://doi.org/10.1063/1.1665472).DOI
  58. M. Stoller, Fourier interpolation from spheres , Trans. Amer. Math. Soc. 374 (2021), 8045--8079. doi:10.1090/tran/8440 (https://doi.org/10.1090/tran/8440).DOI
  59. O. Szehr and R. Zarouf, Projected inner-function dynamics and crystalline measures of Meyer--Blaschke type , preprint, arXiv:2607.19026 (https://arxiv.org/abs/2607.19026) (2026).arXiv
  60. A. Tselishchev, A new example of crystalline-type measure with unit masses and an application to tiling of the real line by translates of a function , preprint, arXiv:2608.09354 (https://arxiv.org/abs/2608.09354) (2026).arXiv
  61. M. Viazovska, Sharp sphere packings , in: Proceedings of the International Congress of Mathematicians (Rio de Janeiro, 2018), Vol. II, World Sci. Publ., Hackensack, NJ, 2018, 455--466. doi:10.1142/9789813272880_0063 (https://doi.org/10.1142/9789813272880_0063).DOI
  62. M. Viazovska, On discrete Fourier uniqueness sets in Euclidean space , in: Proceedings of the International Congress of Mathematicians 2022, Vol. 1, EMS Press, Berlin, 2023, 270--283. doi:10.4171/icm2022/207 (https://doi.org/10.4171/icm2022/207).DOI

Version history

  1. v1 · 57 pagesSubmitted by Jamie MartinInitial depositCurrentNov 10, 2026