--- _id: '14245' abstract: - lang: eng text: We establish effective counting results for lattice points in families of domains in real, complex and quaternionic hyperbolic spaces of any dimension. The domains we focus on are defined as product sets with respect to an Iwasawa decomposition. Several natural diophantine problems can be reduced to counting lattice points in such domains. These include equidistribution of the ratio of the length of the shortest solution (x,y) to the gcd equation bx−ay=1 relative to the length of (a,b), where (a,b) ranges over primitive vectors in a disc whose radius increases, the natural analog of this problem in imaginary quadratic number fields, as well as equidistribution of integral solutions to the diophantine equation defined by an integral Lorentz form in three or more variables. We establish an effective rate of convergence for these equidistribution problems, depending on the size of the spectral gap associated with a suitable lattice subgroup in the isometry group of the relevant hyperbolic space. The main result underlying our discussion amounts to establishing effective joint equidistribution for the horospherical component and the radial component in the Iwasawa decomposition of lattice elements. acknowledgement: The authors thank the referee for important comments which led to significant improvements is the presentation of several results in the paper. They also thank Ami Paz for preparing the figures for this paper. Horesh thanks Ami Paz and Yakov Karasik for helpful discussions. Nevo thanks John Parker and Rene Rühr for providing some very useful references. Nevo is supported by ISF Grant No. 2095/15. article_processing_charge: Yes article_type: original author: - first_name: Tal full_name: Horesh, Tal id: C8B7BF48-8D81-11E9-BCA9-F536E6697425 last_name: Horesh - first_name: Amos full_name: Nevo, Amos last_name: Nevo citation: ama: 'Horesh T, Nevo A. Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution. Pacific Journal of Mathematics. 2023;324(2):265-294. doi:10.2140/pjm.2023.324.265' apa: 'Horesh, T., & Nevo, A. (2023). Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution. Pacific Journal of Mathematics. Mathematical Sciences Publishers. https://doi.org/10.2140/pjm.2023.324.265' chicago: 'Horesh, Tal, and Amos Nevo. “Horospherical Coordinates of Lattice Points in Hyperbolic Spaces: Effective Counting and Equidistribution.” Pacific Journal of Mathematics. Mathematical Sciences Publishers, 2023. https://doi.org/10.2140/pjm.2023.324.265.' ieee: 'T. Horesh and A. Nevo, “Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution,” Pacific Journal of Mathematics, vol. 324, no. 2. Mathematical Sciences Publishers, pp. 265–294, 2023.' ista: 'Horesh T, Nevo A. 2023. Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution. Pacific Journal of Mathematics. 324(2), 265–294.' mla: 'Horesh, Tal, and Amos Nevo. “Horospherical Coordinates of Lattice Points in Hyperbolic Spaces: Effective Counting and Equidistribution.” Pacific Journal of Mathematics, vol. 324, no. 2, Mathematical Sciences Publishers, 2023, pp. 265–94, doi:10.2140/pjm.2023.324.265.' short: T. Horesh, A. Nevo, Pacific Journal of Mathematics 324 (2023) 265–294. date_created: 2023-08-27T22:01:18Z date_published: 2023-07-26T00:00:00Z date_updated: 2023-12-13T12:19:42Z day: '26' ddc: - '510' department: - _id: TiBr doi: 10.2140/pjm.2023.324.265 external_id: arxiv: - '1612.08215' isi: - '001047690500001' file: - access_level: open_access checksum: a675b53cfb31fa46be1e879b7e77fe8c content_type: application/pdf creator: dernst date_created: 2023-09-05T07:26:17Z date_updated: 2023-09-05T07:26:17Z file_id: '14267' file_name: 2023_PacificJourMaths_Horesh.pdf file_size: 654895 relation: main_file success: 1 file_date_updated: 2023-09-05T07:26:17Z has_accepted_license: '1' intvolume: ' 324' isi: 1 issue: '2' language: - iso: eng month: '07' oa: 1 oa_version: Published Version page: 265-294 publication: Pacific Journal of Mathematics publication_identifier: eissn: - 1945-5844 issn: - 0030-8730 publication_status: published publisher: Mathematical Sciences Publishers quality_controlled: '1' scopus_import: '1' status: public title: 'Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution' tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 324 year: '2023' ... --- _id: '14717' abstract: - lang: eng text: We count primitive lattices of rank d inside Zn as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subspace that a lattice spans, namely its projection to the Grassmannian; its homothety class and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets in the spaces of parameters that are general enough to conclude the joint equidistribution of these parameters. In addition to the primitive d-lattices Λ themselves, we also consider their orthogonal complements in Zn⁠, A1⁠, and show that the equidistribution occurs jointly for Λ and A1⁠. Finally, our asymptotic formulas for the number of primitive lattices include an explicit bound on the error term. acknowledgement: This work was done when both authors were visiting Institute of Science and Technology (IST) Austria. T.H. was being supported by Engineering and Physical Sciences Research Council grant EP/P026710/1. Y.K. had a great time there and is grateful for the hospitality. The appendix to this paper is largely based on a mini course T.H. had given at IST in February 2020. article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Tal full_name: Horesh, Tal id: C8B7BF48-8D81-11E9-BCA9-F536E6697425 last_name: Horesh - first_name: Yakov full_name: Karasik, Yakov last_name: Karasik citation: ama: Horesh T, Karasik Y. Equidistribution of primitive lattices in ℝn. Quarterly Journal of Mathematics. 2023;74(4):1253-1294. doi:10.1093/qmath/haad008 apa: Horesh, T., & Karasik, Y. (2023). Equidistribution of primitive lattices in ℝn. Quarterly Journal of Mathematics. Oxford University Press. https://doi.org/10.1093/qmath/haad008 chicago: Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices in ℝn.” Quarterly Journal of Mathematics. Oxford University Press, 2023. https://doi.org/10.1093/qmath/haad008. ieee: T. Horesh and Y. Karasik, “Equidistribution of primitive lattices in ℝn,” Quarterly Journal of Mathematics, vol. 74, no. 4. Oxford University Press, pp. 1253–1294, 2023. ista: Horesh T, Karasik Y. 2023. Equidistribution of primitive lattices in ℝn. Quarterly Journal of Mathematics. 74(4), 1253–1294. mla: Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices in ℝn.” Quarterly Journal of Mathematics, vol. 74, no. 4, Oxford University Press, 2023, pp. 1253–94, doi:10.1093/qmath/haad008. short: T. Horesh, Y. Karasik, Quarterly Journal of Mathematics 74 (2023) 1253–1294. date_created: 2023-12-31T23:01:03Z date_published: 2023-12-01T00:00:00Z date_updated: 2024-01-02T07:39:55Z day: '01' ddc: - '510' department: - _id: TiBr doi: 10.1093/qmath/haad008 external_id: arxiv: - '2012.04508' file: - access_level: open_access checksum: bf29baa9eae8500f3374dbcb80712687 content_type: application/pdf creator: dernst date_created: 2024-01-02T07:37:09Z date_updated: 2024-01-02T07:37:09Z file_id: '14720' file_name: 2023_QuarterlyJourMath_Horesh.pdf file_size: 724748 relation: main_file success: 1 file_date_updated: 2024-01-02T07:37:09Z has_accepted_license: '1' intvolume: ' 74' issue: '4' language: - iso: eng month: '12' oa: 1 oa_version: Published Version page: 1253-1294 project: - _id: 26A8D266-B435-11E9-9278-68D0E5697425 grant_number: EP-P026710-2 name: Between rational and integral points publication: Quarterly Journal of Mathematics publication_identifier: eissn: - 1464-3847 issn: - 0033-5606 publication_status: published publisher: Oxford University Press quality_controlled: '1' scopus_import: '1' status: public title: Equidistribution of primitive lattices in ℝn tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 74 year: '2023' ... --- _id: '9199' abstract: - lang: eng text: "We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's programme on \"freeness\" for rational points of bounded height on Fano\r\nvarieties." acknowledgement: The authors are very grateful to Will Sawin for useful remarks about this topic. While working on this paper the first two authors were supported by EPSRC grant EP/P026710/1, and the first and last authors by FWF grant P 32428-N35. article_processing_charge: No article_type: original author: - first_name: Timothy D full_name: Browning, Timothy D id: 35827D50-F248-11E8-B48F-1D18A9856A87 last_name: Browning orcid: 0000-0002-8314-0177 - first_name: Tal full_name: Horesh, Tal id: C8B7BF48-8D81-11E9-BCA9-F536E6697425 last_name: Horesh - first_name: Florian Alexander full_name: Wilsch, Florian Alexander id: 560601DA-8D36-11E9-A136-7AC1E5697425 last_name: Wilsch orcid: 0000-0001-7302-8256 citation: ama: Browning TD, Horesh T, Wilsch FA. Equidistribution and freeness on Grassmannians. Algebra & Number Theory. 2022;16(10):2385-2407. doi:10.2140/ant.2022.16.2385 apa: Browning, T. D., Horesh, T., & Wilsch, F. A. (2022). Equidistribution and freeness on Grassmannians. Algebra & Number Theory. Mathematical Sciences Publishers. https://doi.org/10.2140/ant.2022.16.2385 chicago: Browning, Timothy D, Tal Horesh, and Florian Alexander Wilsch. “Equidistribution and Freeness on Grassmannians.” Algebra & Number Theory. Mathematical Sciences Publishers, 2022. https://doi.org/10.2140/ant.2022.16.2385. ieee: T. D. Browning, T. Horesh, and F. A. Wilsch, “Equidistribution and freeness on Grassmannians,” Algebra & Number Theory, vol. 16, no. 10. Mathematical Sciences Publishers, pp. 2385–2407, 2022. ista: Browning TD, Horesh T, Wilsch FA. 2022. Equidistribution and freeness on Grassmannians. Algebra & Number Theory. 16(10), 2385–2407. mla: Browning, Timothy D., et al. “Equidistribution and Freeness on Grassmannians.” Algebra & Number Theory, vol. 16, no. 10, Mathematical Sciences Publishers, 2022, pp. 2385–407, doi:10.2140/ant.2022.16.2385. short: T.D. Browning, T. Horesh, F.A. Wilsch, Algebra & Number Theory 16 (2022) 2385–2407. date_created: 2021-02-25T09:56:57Z date_published: 2022-12-01T00:00:00Z date_updated: 2023-08-02T06:46:38Z day: '01' department: - _id: TiBr doi: 10.2140/ant.2022.16.2385 external_id: arxiv: - '2102.11552' isi: - '000961514100004' intvolume: ' 16' isi: 1 issue: '10' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2102.11552 month: '12' oa: 1 oa_version: Preprint page: 2385-2407 project: - _id: 26A8D266-B435-11E9-9278-68D0E5697425 grant_number: EP-P026710-2 name: Between rational and integral points - _id: 26AEDAB2-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: P32428 name: New frontiers of the Manin conjecture publication: Algebra & Number Theory publication_identifier: eissn: - 1944-7833 issn: - 1937-0652 publication_status: published publisher: Mathematical Sciences Publishers quality_controlled: '1' scopus_import: '1' status: public title: Equidistribution and freeness on Grassmannians type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 16 year: '2022' ... --- _id: '12684' abstract: - lang: eng text: Given a place ω of a global function field K over a finite field, with associated affine function ring Rω and completion Kω , the aim of this paper is to give an effective joint equidistribution result for renormalized primitive lattice points (a,b)∈Rω2 in the plane Kω2 , and for renormalized solutions to the gcd equation ax+by=1 . The main tools are techniques of Goronik and Nevo for counting lattice points in well-rounded families of subsets. This gives a sharper analog in positive characteristic of a result of Nevo and the first author for the equidistribution of the primitive lattice points in \ZZ2 . acknowledgement: "The authors warmly thank Amos Nevo for having presented the authors to each other during\r\na beautiful conference in Goa in February 2016, where the idea of this paper was born. The\r\nfirst author thanks the IHES for two post-doctoral years when most of this paper was discussed,\r\nand the Topology team in Orsay for financial support at the final stage. The first author was\r\nsupported by the EPRSC EP/P026710/1 grant. Finally, we warmly thank the referee for many\r\nvery helpful comments that have improved the readability of this paper." article_processing_charge: No article_type: original author: - first_name: Tal full_name: Horesh, Tal id: C8B7BF48-8D81-11E9-BCA9-F536E6697425 last_name: Horesh - first_name: Frédéric full_name: Paulin, Frédéric last_name: Paulin citation: ama: Horesh T, Paulin F. Effective equidistribution of lattice points in positive characteristic. Journal de Theorie des Nombres de Bordeaux. 2022;34(3):679-703. doi:10.5802/JTNB.1222 apa: Horesh, T., & Paulin, F. (2022). Effective equidistribution of lattice points in positive characteristic. Journal de Theorie Des Nombres de Bordeaux. Centre Mersenne. https://doi.org/10.5802/JTNB.1222 chicago: Horesh, Tal, and Frédéric Paulin. “Effective Equidistribution of Lattice Points in Positive Characteristic.” Journal de Theorie Des Nombres de Bordeaux. Centre Mersenne, 2022. https://doi.org/10.5802/JTNB.1222. ieee: T. Horesh and F. Paulin, “Effective equidistribution of lattice points in positive characteristic,” Journal de Theorie des Nombres de Bordeaux, vol. 34, no. 3. Centre Mersenne, pp. 679–703, 2022. ista: Horesh T, Paulin F. 2022. Effective equidistribution of lattice points in positive characteristic. Journal de Theorie des Nombres de Bordeaux. 34(3), 679–703. mla: Horesh, Tal, and Frédéric Paulin. “Effective Equidistribution of Lattice Points in Positive Characteristic.” Journal de Theorie Des Nombres de Bordeaux, vol. 34, no. 3, Centre Mersenne, 2022, pp. 679–703, doi:10.5802/JTNB.1222. short: T. Horesh, F. Paulin, Journal de Theorie Des Nombres de Bordeaux 34 (2022) 679–703. date_created: 2023-02-26T23:01:02Z date_published: 2022-01-27T00:00:00Z date_updated: 2023-08-04T10:41:40Z day: '27' ddc: - '510' department: - _id: TiBr doi: 10.5802/JTNB.1222 external_id: arxiv: - '2001.01534' isi: - '000926504300003' file: - access_level: open_access checksum: 08f28fded270251f568f610cf5166d69 content_type: application/pdf creator: dernst date_created: 2023-02-27T09:10:13Z date_updated: 2023-02-27T09:10:13Z file_id: '12689' file_name: 2023_JourTheorieNombreBordeaux_Horesh.pdf file_size: 870468 relation: main_file success: 1 file_date_updated: 2023-02-27T09:10:13Z has_accepted_license: '1' intvolume: ' 34' isi: 1 issue: '3' language: - iso: eng license: https://creativecommons.org/licenses/by-nd/4.0/ month: '01' oa: 1 oa_version: Published Version page: 679-703 publication: Journal de Theorie des Nombres de Bordeaux publication_identifier: eissn: - 2118-8572 issn: - 1246-7405 publication_status: published publisher: Centre Mersenne quality_controlled: '1' scopus_import: '1' status: public title: Effective equidistribution of lattice points in positive characteristic tmp: image: /image/cc_by_nd.png legal_code_url: https://creativecommons.org/licenses/by-nd/4.0/legalcode name: Creative Commons Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0) short: CC BY-ND (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 34 year: '2022' ...