[{"quality_controlled":"1","ddc":["510"],"page":"1253-1294","type":"journal_article","date_updated":"2024-01-02T07:39:55Z","_id":"14717","publisher":"Oxford University Press","doi":"10.1093/qmath/haad008","article_processing_charge":"Yes (via OA deal)","date_published":"2023-12-01T00:00:00Z","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.","project":[{"_id":"26A8D266-B435-11E9-9278-68D0E5697425","name":"Between rational and integral points","grant_number":"EP-P026710-2"}],"publication":"Quarterly Journal of Mathematics","status":"public","external_id":{"arxiv":["2012.04508"]},"year":"2023","publication_status":"published","publication_identifier":{"issn":["0033-5606"],"eissn":["1464-3847"]},"file_date_updated":"2024-01-02T07:37:09Z","intvolume":"        74","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"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."}],"has_accepted_license":"1","article_type":"original","date_created":"2023-12-31T23:01:03Z","volume":74,"oa_version":"Published Version","title":"Equidistribution of primitive lattices in ℝn","author":[{"first_name":"Tal","id":"C8B7BF48-8D81-11E9-BCA9-F536E6697425","full_name":"Horesh, Tal","last_name":"Horesh"},{"full_name":"Karasik, Yakov","last_name":"Karasik","first_name":"Yakov"}],"day":"01","scopus_import":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","issue":"4","citation":{"ieee":"T. Horesh and Y. Karasik, “Equidistribution of primitive lattices in ℝn,” <i>Quarterly Journal of Mathematics</i>, vol. 74, no. 4. Oxford University Press, pp. 1253–1294, 2023.","short":"T. Horesh, Y. Karasik, Quarterly Journal of Mathematics 74 (2023) 1253–1294.","ama":"Horesh T, Karasik Y. Equidistribution of primitive lattices in ℝn. <i>Quarterly Journal of Mathematics</i>. 2023;74(4):1253-1294. doi:<a href=\"https://doi.org/10.1093/qmath/haad008\">10.1093/qmath/haad008</a>","apa":"Horesh, T., &#38; Karasik, Y. (2023). Equidistribution of primitive lattices in ℝn. <i>Quarterly Journal of Mathematics</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/qmath/haad008\">https://doi.org/10.1093/qmath/haad008</a>","mla":"Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices in ℝn.” <i>Quarterly Journal of Mathematics</i>, vol. 74, no. 4, Oxford University Press, 2023, pp. 1253–94, doi:<a href=\"https://doi.org/10.1093/qmath/haad008\">10.1093/qmath/haad008</a>.","ista":"Horesh T, Karasik Y. 2023. Equidistribution of primitive lattices in ℝn. Quarterly Journal of Mathematics. 74(4), 1253–1294.","chicago":"Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices in ℝn.” <i>Quarterly Journal of Mathematics</i>. Oxford University Press, 2023. <a href=\"https://doi.org/10.1093/qmath/haad008\">https://doi.org/10.1093/qmath/haad008</a>."},"language":[{"iso":"eng"}],"oa":1,"file":[{"date_created":"2024-01-02T07:37:09Z","file_size":724748,"date_updated":"2024-01-02T07:37:09Z","creator":"dernst","file_id":"14720","file_name":"2023_QuarterlyJourMath_Horesh.pdf","success":1,"content_type":"application/pdf","access_level":"open_access","relation":"main_file","checksum":"bf29baa9eae8500f3374dbcb80712687"}],"department":[{"_id":"TiBr"}],"month":"12","arxiv":1},{"status":"public","publication":"Pacific Journal of Mathematics","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.","date_published":"2023-07-26T00:00:00Z","isi":1,"year":"2023","external_id":{"arxiv":["1612.08215"],"isi":["001047690500001"]},"page":"265-294","ddc":["510"],"quality_controlled":"1","article_processing_charge":"Yes","doi":"10.2140/pjm.2023.324.265","publisher":"Mathematical Sciences Publishers","_id":"14245","date_updated":"2023-12-13T12:19:42Z","type":"journal_article","oa":1,"language":[{"iso":"eng"}],"citation":{"mla":"Horesh, Tal, and Amos Nevo. “Horospherical Coordinates of Lattice Points in Hyperbolic Spaces: Effective Counting and Equidistribution.” <i>Pacific Journal of Mathematics</i>, vol. 324, no. 2, Mathematical Sciences Publishers, 2023, pp. 265–94, doi:<a href=\"https://doi.org/10.2140/pjm.2023.324.265\">10.2140/pjm.2023.324.265</a>.","apa":"Horesh, T., &#38; Nevo, A. (2023). Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution. <i>Pacific Journal of Mathematics</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/pjm.2023.324.265\">https://doi.org/10.2140/pjm.2023.324.265</a>","chicago":"Horesh, Tal, and Amos Nevo. “Horospherical Coordinates of Lattice Points in Hyperbolic Spaces: Effective Counting and Equidistribution.” <i>Pacific Journal of Mathematics</i>. Mathematical Sciences Publishers, 2023. <a href=\"https://doi.org/10.2140/pjm.2023.324.265\">https://doi.org/10.2140/pjm.2023.324.265</a>.","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.","short":"T. Horesh, A. Nevo, Pacific Journal of Mathematics 324 (2023) 265–294.","ieee":"T. Horesh and A. Nevo, “Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution,” <i>Pacific Journal of Mathematics</i>, vol. 324, no. 2. Mathematical Sciences Publishers, pp. 265–294, 2023.","ama":"Horesh T, Nevo A. Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution. <i>Pacific Journal of Mathematics</i>. 2023;324(2):265-294. doi:<a href=\"https://doi.org/10.2140/pjm.2023.324.265\">10.2140/pjm.2023.324.265</a>"},"issue":"2","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","arxiv":1,"month":"07","department":[{"_id":"TiBr"}],"file":[{"file_name":"2023_PacificJourMaths_Horesh.pdf","success":1,"content_type":"application/pdf","access_level":"open_access","relation":"main_file","checksum":"a675b53cfb31fa46be1e879b7e77fe8c","file_size":654895,"date_created":"2023-09-05T07:26:17Z","creator":"dernst","date_updated":"2023-09-05T07:26:17Z","file_id":"14267"}],"has_accepted_license":"1","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."}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"intvolume":"       324","file_date_updated":"2023-09-05T07:26:17Z","publication_identifier":{"eissn":["1945-5844"],"issn":["0030-8730"]},"publication_status":"published","day":"26","scopus_import":"1","author":[{"id":"C8B7BF48-8D81-11E9-BCA9-F536E6697425","full_name":"Horesh, Tal","last_name":"Horesh","first_name":"Tal"},{"last_name":"Nevo","full_name":"Nevo, Amos","first_name":"Amos"}],"title":"Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution","oa_version":"Published Version","volume":324,"date_created":"2023-08-27T22:01:18Z","article_type":"original"},{"department":[{"_id":"TiBr"}],"month":"12","arxiv":1,"citation":{"chicago":"Browning, Timothy D, Tal Horesh, and Florian Alexander Wilsch. “Equidistribution and Freeness on Grassmannians.” <i>Algebra &#38; Number Theory</i>. Mathematical Sciences Publishers, 2022. <a href=\"https://doi.org/10.2140/ant.2022.16.2385\">https://doi.org/10.2140/ant.2022.16.2385</a>.","ista":"Browning TD, Horesh T, Wilsch FA. 2022. Equidistribution and freeness on Grassmannians. Algebra &#38; Number Theory. 16(10), 2385–2407.","mla":"Browning, Timothy D., et al. “Equidistribution and Freeness on Grassmannians.” <i>Algebra &#38; Number Theory</i>, vol. 16, no. 10, Mathematical Sciences Publishers, 2022, pp. 2385–407, doi:<a href=\"https://doi.org/10.2140/ant.2022.16.2385\">10.2140/ant.2022.16.2385</a>.","apa":"Browning, T. D., Horesh, T., &#38; Wilsch, F. A. (2022). Equidistribution and freeness on Grassmannians. <i>Algebra &#38; Number Theory</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/ant.2022.16.2385\">https://doi.org/10.2140/ant.2022.16.2385</a>","ama":"Browning TD, Horesh T, Wilsch FA. Equidistribution and freeness on Grassmannians. <i>Algebra &#38; Number Theory</i>. 2022;16(10):2385-2407. doi:<a href=\"https://doi.org/10.2140/ant.2022.16.2385\">10.2140/ant.2022.16.2385</a>","short":"T.D. Browning, T. Horesh, F.A. Wilsch, Algebra &#38; Number Theory 16 (2022) 2385–2407.","ieee":"T. D. Browning, T. Horesh, and F. A. Wilsch, “Equidistribution and freeness on Grassmannians,” <i>Algebra &#38; Number Theory</i>, vol. 16, no. 10. Mathematical Sciences Publishers, pp. 2385–2407, 2022."},"issue":"10","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa":1,"language":[{"iso":"eng"}],"volume":16,"date_created":"2021-02-25T09:56:57Z","article_type":"original","scopus_import":"1","day":"01","author":[{"orcid":"0000-0002-8314-0177","first_name":"Timothy D","last_name":"Browning","id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D"},{"last_name":"Horesh","full_name":"Horesh, Tal","id":"C8B7BF48-8D81-11E9-BCA9-F536E6697425","first_name":"Tal"},{"first_name":"Florian Alexander","orcid":"0000-0001-7302-8256","last_name":"Wilsch","id":"560601DA-8D36-11E9-A136-7AC1E5697425","full_name":"Wilsch, Florian Alexander"}],"oa_version":"Preprint","title":"Equidistribution and freeness on Grassmannians","publication_identifier":{"issn":["1937-0652"],"eissn":["1944-7833"]},"publication_status":"published","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."}],"intvolume":"        16","year":"2022","isi":1,"external_id":{"isi":["000961514100004"],"arxiv":["2102.11552"]},"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.","date_published":"2022-12-01T00:00:00Z","publication":"Algebra & Number Theory","status":"public","project":[{"name":"Between rational and integral points","grant_number":"EP-P026710-2","_id":"26A8D266-B435-11E9-9278-68D0E5697425"},{"_id":"26AEDAB2-B435-11E9-9278-68D0E5697425","grant_number":"P32428","name":"New frontiers of the Manin conjecture","call_identifier":"FWF"}],"_id":"9199","date_updated":"2023-08-02T06:46:38Z","type":"journal_article","article_processing_charge":"No","doi":"10.2140/ant.2022.16.2385","publisher":"Mathematical Sciences Publishers","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2102.11552"}],"quality_controlled":"1","page":"2385-2407"},{"language":[{"iso":"eng"}],"oa":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","issue":"3","citation":{"ama":"Horesh T, Paulin F. Effective equidistribution of lattice points in positive characteristic. <i>Journal de Theorie des Nombres de Bordeaux</i>. 2022;34(3):679-703. doi:<a href=\"https://doi.org/10.5802/JTNB.1222\">10.5802/JTNB.1222</a>","ieee":"T. Horesh and F. Paulin, “Effective equidistribution of lattice points in positive characteristic,” <i>Journal de Theorie des Nombres de Bordeaux</i>, vol. 34, no. 3. Centre Mersenne, pp. 679–703, 2022.","short":"T. Horesh, F. Paulin, Journal de Theorie Des Nombres de Bordeaux 34 (2022) 679–703.","chicago":"Horesh, Tal, and Frédéric Paulin. “Effective Equidistribution of Lattice Points in Positive Characteristic.” <i>Journal de Theorie Des Nombres de Bordeaux</i>. Centre Mersenne, 2022. <a href=\"https://doi.org/10.5802/JTNB.1222\">https://doi.org/10.5802/JTNB.1222</a>.","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.","apa":"Horesh, T., &#38; Paulin, F. (2022). Effective equidistribution of lattice points in positive characteristic. <i>Journal de Theorie Des Nombres de Bordeaux</i>. Centre Mersenne. <a href=\"https://doi.org/10.5802/JTNB.1222\">https://doi.org/10.5802/JTNB.1222</a>","mla":"Horesh, Tal, and Frédéric Paulin. “Effective Equidistribution of Lattice Points in Positive Characteristic.” <i>Journal de Theorie Des Nombres de Bordeaux</i>, vol. 34, no. 3, Centre Mersenne, 2022, pp. 679–703, doi:<a href=\"https://doi.org/10.5802/JTNB.1222\">10.5802/JTNB.1222</a>."},"arxiv":1,"month":"01","file":[{"content_type":"application/pdf","access_level":"open_access","file_name":"2023_JourTheorieNombreBordeaux_Horesh.pdf","success":1,"checksum":"08f28fded270251f568f610cf5166d69","relation":"main_file","creator":"dernst","date_updated":"2023-02-27T09:10:13Z","file_size":870468,"date_created":"2023-02-27T09:10:13Z","file_id":"12689"}],"department":[{"_id":"TiBr"}],"license":"https://creativecommons.org/licenses/by-nd/4.0/","tmp":{"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)","image":"/image/cc_by_nd.png","short":"CC BY-ND (4.0)"},"abstract":[{"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 .","lang":"eng"}],"intvolume":"        34","has_accepted_license":"1","publication_status":"published","publication_identifier":{"eissn":["2118-8572"],"issn":["1246-7405"]},"file_date_updated":"2023-02-27T09:10:13Z","title":"Effective equidistribution of lattice points in positive characteristic","oa_version":"Published Version","author":[{"first_name":"Tal","id":"C8B7BF48-8D81-11E9-BCA9-F536E6697425","full_name":"Horesh, Tal","last_name":"Horesh"},{"first_name":"Frédéric","last_name":"Paulin","full_name":"Paulin, Frédéric"}],"scopus_import":"1","day":"27","article_type":"original","date_created":"2023-02-26T23:01:02Z","volume":34,"status":"public","publication":"Journal de Theorie des Nombres de Bordeaux","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.","date_published":"2022-01-27T00:00:00Z","external_id":{"arxiv":["2001.01534"],"isi":["000926504300003"]},"year":"2022","isi":1,"ddc":["510"],"page":"679-703","quality_controlled":"1","publisher":"Centre Mersenne","doi":"10.5802/JTNB.1222","article_processing_charge":"No","type":"journal_article","date_updated":"2023-08-04T10:41:40Z","_id":"12684"}]