## Poisson approximation and Weibull asymptotics in the geometry of numbers

HTML articles powered by AMS MathViewer

- by Michael Björklund and Alexander Gorodnik;
- Trans. Amer. Math. Soc.
**376**(2023), 2155-2180 - DOI: https://doi.org/10.1090/tran/8826
- Published electronically: December 8, 2022
- PDF | Request permission

## Abstract:

Minkowski’s First Theorem and Dirichlet’s Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some technical conditions, that they exhibit Weibull asymptotics with respect to different natural measures on the space of unimodular lattices in $\mathbb {R}^d$. This follows from very general Poisson approximation results for shrinking targets which should be of independent interest. Furthermore, we show in the appendix that the logarithm laws of Kleinbock-Margulis [Invent. Math. 138 (1999), pp. 451–494], Khinchin and Gallagher [J. London Math. Soc. 37 (1962), pp. 387–390] can be deduced from our distributional results.## References

- Jayadev S. Athreya and Gregory A. Margulis,
*Logarithm laws for unipotent flows. I*, J. Mod. Dyn.**3**(2009), no. 3, 359–378. MR**2538473**, DOI 10.3934/jmd.2009.3.359 - Patrick Billingsley,
*Probability and measure*, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR**1324786** - Michael Björklund and Alexander Gorodnik,
*Central limit theorems for Diophantine approximants*, Math. Ann.**374**(2019), no. 3-4, 1371–1437. MR**3985114**, DOI 10.1007/s00208-019-01828-1 - M. Björklund and A. Gorodnik,
*Counting in generic lattices and higher rank actions*, arXiv:2101.04931, 2021. - M. Björklund and A. Gorodnik,
*Effective multiple equidistribution of translated measures*, To appear in IMRN, arXiv:2105.05468, 2021. - Michael Björklund, Manfred Einsiedler, and Alexander Gorodnik,
*Quantitative multiple mixing*, J. Eur. Math. Soc. (JEMS)**22**(2020), no. 5, 1475–1529. MR**4081727**, DOI 10.4171/jems/949 - J. W. S. Cassels,
*An introduction to the geometry of numbers*, Classics in Mathematics, Springer-Verlag, Berlin, 1997. Corrected reprint of the 1971 edition. MR**1434478** - H. Davenport and W. M. Schmidt,
*Dirichlet’s theorem on diophantine approximation. II*, Acta Arith.**16**(1969/70), 413–424. MR**279040**, DOI 10.4064/aa-16-4-413-424 - D. Dolgopyat, B. Fayad, and L. Sixu,
*Multiple Borel Cantelli Lemma in dynamics and MultiLog law for recurrence*, arXiv:2103.08382, 2021. - P. Gallagher,
*Metric simultaneous diophantine approximation*, J. London Math. Soc.**37**(1962), 387–390. MR**157939**, DOI 10.1112/jlms/s1-37.1.387 - Michael Greenblatt,
*Resolution of singularities, asymptotic expansions of integrals and related phenomena*, J. Anal. Math.**111**(2010), 221–245. MR**2747065**, DOI 10.1007/s11854-010-0016-1 - P. M. Gruber and C. G. Lekkerkerker,
*Geometry of numbers*, 2nd ed., North-Holland Mathematical Library, vol. 37, North-Holland Publishing Co., Amsterdam, 1987. MR**893813** - Masaki Hirata,
*Poisson law for Axiom A diffeomorphisms*, Ergodic Theory Dynam. Systems**13**(1993), no. 3, 533–556. MR**1245828**, DOI 10.1017/S0143385700007513 - Dubi Kelmer and Shucheng Yu,
*The second moment of the Siegel transform in the space of symplectic lattices*, Int. Math. Res. Not. IMRN**8**(2021), 5825–5859. MR**4251265**, DOI 10.1093/imrn/rnz027 - Maxim Sølund Kirsebom,
*Extreme value distributions for one-parameter actions on homogeneous spaces*, Nonlinearity**33**(2020), no. 3, 1218–1239. MR**4063963**, DOI 10.1088/1361-6544/ab5c0c - D. Y. Kleinbock and G. A. Margulis,
*Logarithm laws for flows on homogeneous spaces*, Invent. Math.**138**(1999), no. 3, 451–494. MR**1719827**, DOI 10.1007/s002220050350 - Mark Pollicott,
*Limiting distributions for geodesics excursions on the modular surface*, Spectral analysis in geometry and number theory, Contemp. Math., vol. 484, Amer. Math. Soc., Providence, RI, 2009, pp. 177–185. MR**1500147**, DOI 10.1090/conm/484/09474 - C. A. Rogers,
*Mean values over the space of lattices*, Acta Math.**94**(1955), 249–287. MR**75243**, DOI 10.1007/BF02392493 - Sheldon Ross,
*A first course in probability*, 2nd ed., Macmillan Co., New York; Collier Macmillan Ltd., London, 1984. MR**732623** - Carl Ludwig Siegel,
*Lectures on the geometry of numbers*, Springer-Verlag, Berlin, 1989. Notes by B. Friedman; Rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter; With a preface by Chandrasekharan. MR**1020761**, DOI 10.1007/978-3-662-08287-4 - Carl Ludwig Siegel,
*A mean value theorem in geometry of numbers*, Ann. of Math. (2)**46**(1945), 340–347. MR**12093**, DOI 10.2307/1969027 - Ya. G. Sinaĭ,
*Some mathematical problems in the theory of quantum chaos*, Phys. A**163**(1990), no. 1, 197–204. Statistical physics (Rio de Janeiro, 1989). MR**1043648**, DOI 10.1016/0378-4371(90)90329-Q - Ya. G. Sinaĭ,
*Mathematical problems in the theory of quantum chaos*, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 41–59. MR**1122612**, DOI 10.1007/BFb0089214 - Dennis Sullivan,
*Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics*, Acta Math.**149**(1982), no. 3-4, 215–237. MR**688349**, DOI 10.1007/BF02392354

## Bibliographic Information

**Michael Björklund**- Affiliation: Mathematical Sciences, Chalmers, Chalmers Tvärgata 3, 412 58 Gothenburg, Sweden
- Email: micbjo@chalmers.se
**Alexander Gorodnik**- Affiliation: Department of Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057, Switzerland
- MR Author ID: 649826
- Email: alexander.gorodnik@math.uzh.ch
- Received by editor(s): January 14, 2022
- Received by editor(s) in revised form: September 15, 2022, and September 15, 2022
- Published electronically: December 8, 2022
- Additional Notes: The first author was supported by GoCas Young Excellence grant 11423310 and Swedish VR-grant 11253320, the second author was supported by the SNF grant 200021–182089.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 2155-2180 - MSC (2020): Primary 37A50; Secondary 37A25, 60G70
- DOI: https://doi.org/10.1090/tran/8826
- MathSciNet review: 4549702