[{"has_accepted_license":"1","year":"2023","oa_version":"Published Version","article_type":"original","publication_identifier":{"eissn":["1464-3847"],"issn":["0033-5606"]},"scopus_import":"1","external_id":{"arxiv":["2012.04508"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2024-01-02T07:39:55Z","_id":"14717","abstract":[{"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.","lang":"eng"}],"date_published":"2023-12-01T00:00:00Z","file":[{"content_type":"application/pdf","relation":"main_file","file_id":"14720","success":1,"date_created":"2024-01-02T07:37:09Z","file_name":"2023_QuarterlyJourMath_Horesh.pdf","creator":"dernst","file_size":724748,"date_updated":"2024-01-02T07:37:09Z","access_level":"open_access","checksum":"bf29baa9eae8500f3374dbcb80712687"}],"article_processing_charge":"Yes (via OA deal)","issue":"4","arxiv":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"volume":74,"file_date_updated":"2024-01-02T07:37:09Z","oa":1,"publication_status":"published","type":"journal_article","author":[{"id":"C8B7BF48-8D81-11E9-BCA9-F536E6697425","last_name":"Horesh","first_name":"Tal","full_name":"Horesh, Tal"},{"last_name":"Karasik","full_name":"Karasik, Yakov","first_name":"Yakov"}],"day":"01","title":"Equidistribution of primitive lattices in ℝn","citation":{"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>","short":"T. Horesh, Y. Karasik, Quarterly Journal of Mathematics 74 (2023) 1253–1294.","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>.","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>.","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.","ista":"Horesh T, Karasik Y. 2023. Equidistribution of primitive lattices in ℝn. Quarterly Journal of Mathematics. 74(4), 1253–1294."},"doi":"10.1093/qmath/haad008","ddc":["510"],"language":[{"iso":"eng"}],"project":[{"_id":"26A8D266-B435-11E9-9278-68D0E5697425","grant_number":"EP-P026710-2","name":"Between rational and integral points"}],"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.","date_created":"2023-12-31T23:01:03Z","month":"12","page":"1253-1294","status":"public","intvolume":"        74","publication":"Quarterly Journal of Mathematics","quality_controlled":"1","department":[{"_id":"TiBr"}],"publisher":"Oxford University Press"},{"author":[{"orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","full_name":"Browning, Timothy D","first_name":"Timothy D"},{"full_name":"Sawin, Will","first_name":"Will","last_name":"Sawin"}],"type":"journal_article","day":"12","title":"Free rational curves on low degree hypersurfaces and the circle method","citation":{"mla":"Browning, Timothy D., and Will Sawin. “Free Rational Curves on Low Degree Hypersurfaces and the Circle Method.” <i>Algebra and Number Theory</i>, vol. 17, no. 3, Mathematical Sciences Publishers, 2023, pp. 719–48, doi:<a href=\"https://doi.org/10.2140/ant.2023.17.719\">10.2140/ant.2023.17.719</a>.","ista":"Browning TD, Sawin W. 2023. Free rational curves on low degree hypersurfaces and the circle method. Algebra and Number Theory. 17(3), 719–748.","ieee":"T. D. Browning and W. Sawin, “Free rational curves on low degree hypersurfaces and the circle method,” <i>Algebra and Number Theory</i>, vol. 17, no. 3. Mathematical Sciences Publishers, pp. 719–748, 2023.","chicago":"Browning, Timothy D, and Will Sawin. “Free Rational Curves on Low Degree Hypersurfaces and the Circle Method.” <i>Algebra and Number Theory</i>. Mathematical Sciences Publishers, 2023. <a href=\"https://doi.org/10.2140/ant.2023.17.719\">https://doi.org/10.2140/ant.2023.17.719</a>.","apa":"Browning, T. D., &#38; Sawin, W. (2023). Free rational curves on low degree hypersurfaces and the circle method. <i>Algebra and Number Theory</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/ant.2023.17.719\">https://doi.org/10.2140/ant.2023.17.719</a>","ama":"Browning TD, Sawin W. Free rational curves on low degree hypersurfaces and the circle method. <i>Algebra and Number Theory</i>. 2023;17(3):719-748. doi:<a href=\"https://doi.org/10.2140/ant.2023.17.719\">10.2140/ant.2023.17.719</a>","short":"T.D. Browning, W. Sawin, Algebra and Number Theory 17 (2023) 719–748."},"doi":"10.2140/ant.2023.17.719","ddc":["510"],"language":[{"iso":"eng"}],"project":[{"grant_number":"EP-P026710-2","name":"Between rational and integral points","_id":"26A8D266-B435-11E9-9278-68D0E5697425"}],"acknowledgement":"The authors are grateful to Paul Nelson, Per Salberger and Jason Starr for useful comments. While working on this paper the first author was supported by EPRSC grant EP/P026710/1. The research was partially conducted during the period the second author served as a Clay Research Fellow, and partially conducted during the period he was supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation.","date_created":"2023-05-28T22:01:02Z","month":"04","page":"719-748","status":"public","intvolume":"        17","publication":"Algebra and Number Theory","quality_controlled":"1","department":[{"_id":"TiBr"}],"isi":1,"publisher":"Mathematical Sciences Publishers","year":"2023","has_accepted_license":"1","oa_version":"Published Version","article_type":"original","publication_identifier":{"issn":["1937-0652"],"eissn":["1944-7833"]},"scopus_import":"1","external_id":{"isi":["000996014700004"],"arxiv":["1810.06882"]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_updated":"2023-08-01T14:51:57Z","_id":"13091","abstract":[{"text":"We use a function field version of the Hardy–Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre’s reformulation of the Batyrev–Manin conjecture in terms of slopes with respect to the tangent bundle.","lang":"eng"}],"date_published":"2023-04-12T00:00:00Z","file":[{"access_level":"open_access","date_updated":"2023-05-30T08:05:22Z","checksum":"5d5d67b235905650e33cf7065d7583b4","file_size":1430719,"creator":"dernst","file_name":"2023_AlgebraNumberTheory_Browning.pdf","success":1,"date_created":"2023-05-30T08:05:22Z","relation":"main_file","file_id":"13101","content_type":"application/pdf"}],"article_processing_charge":"No","issue":"3","arxiv":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"volume":17,"file_date_updated":"2023-05-30T08:05:22Z","oa":1,"publication_status":"published"},{"page":"2385-2407","month":"12","date_created":"2021-02-25T09:56:57Z","publisher":"Mathematical Sciences Publishers","isi":1,"department":[{"_id":"TiBr"}],"quality_controlled":"1","publication":"Algebra & Number Theory","status":"public","intvolume":"        16","citation":{"short":"T.D. Browning, T. Horesh, F.A. Wilsch, Algebra &#38; Number Theory 16 (2022) 2385–2407.","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>","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.","ista":"Browning TD, Horesh T, Wilsch FA. 2022. Equidistribution and freeness on Grassmannians. Algebra &#38; Number Theory. 16(10), 2385–2407.","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>.","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>."},"title":"Equidistribution and freeness on Grassmannians","day":"01","author":[{"first_name":"Timothy D","full_name":"Browning, Timothy D","last_name":"Browning","orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87"},{"id":"C8B7BF48-8D81-11E9-BCA9-F536E6697425","last_name":"Horesh","full_name":"Horesh, Tal","first_name":"Tal"},{"orcid":"0000-0001-7302-8256","id":"560601DA-8D36-11E9-A136-7AC1E5697425","last_name":"Wilsch","full_name":"Wilsch, Florian Alexander","first_name":"Florian Alexander"}],"type":"journal_article","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.","project":[{"name":"Between rational and integral points","grant_number":"EP-P026710-2","_id":"26A8D266-B435-11E9-9278-68D0E5697425"},{"grant_number":"P32428","name":"New frontiers of the Manin conjecture","call_identifier":"FWF","_id":"26AEDAB2-B435-11E9-9278-68D0E5697425"}],"language":[{"iso":"eng"}],"doi":"10.2140/ant.2022.16.2385","arxiv":1,"issue":"10","article_processing_charge":"No","abstract":[{"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.","lang":"eng"}],"_id":"9199","date_published":"2022-12-01T00:00:00Z","publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2102.11552"}],"oa":1,"volume":16,"article_type":"original","year":"2022","oa_version":"Preprint","date_updated":"2023-08-02T06:46:38Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","external_id":{"isi":["000961514100004"],"arxiv":["2102.11552"]},"scopus_import":"1","publication_identifier":{"eissn":["1944-7833"],"issn":["1937-0652"]}},{"title":"Arithmetic of higher-dimensional orbifolds and a mixed Waring problem","citation":{"short":"T.D. Browning, S. Yamagishi, Mathematische Zeitschrift 299 (2021) 1071–1101.","apa":"Browning, T. D., &#38; Yamagishi, S. (2021). Arithmetic of higher-dimensional orbifolds and a mixed Waring problem. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-021-02695-w\">https://doi.org/10.1007/s00209-021-02695-w</a>","ama":"Browning TD, Yamagishi S. Arithmetic of higher-dimensional orbifolds and a mixed Waring problem. <i>Mathematische Zeitschrift</i>. 2021;299:1071–1101. doi:<a href=\"https://doi.org/10.1007/s00209-021-02695-w\">10.1007/s00209-021-02695-w</a>","ieee":"T. D. Browning and S. Yamagishi, “Arithmetic of higher-dimensional orbifolds and a mixed Waring problem,” <i>Mathematische Zeitschrift</i>, vol. 299. Springer Nature, pp. 1071–1101, 2021.","ista":"Browning TD, Yamagishi S. 2021. Arithmetic of higher-dimensional orbifolds and a mixed Waring problem. Mathematische Zeitschrift. 299, 1071–1101.","chicago":"Browning, Timothy D, and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional Orbifolds and a Mixed Waring Problem.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00209-021-02695-w\">https://doi.org/10.1007/s00209-021-02695-w</a>.","mla":"Browning, Timothy D., and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional Orbifolds and a Mixed Waring Problem.” <i>Mathematische Zeitschrift</i>, vol. 299, Springer Nature, 2021, pp. 1071–1101, doi:<a href=\"https://doi.org/10.1007/s00209-021-02695-w\">10.1007/s00209-021-02695-w</a>."},"author":[{"last_name":"Browning","first_name":"Timothy D","full_name":"Browning, Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8314-0177"},{"full_name":"Yamagishi, Shuntaro","first_name":"Shuntaro","last_name":"Yamagishi"}],"type":"journal_article","day":"05","project":[{"name":"Between rational and integral points","grant_number":"EP-P026710-2","_id":"26A8D266-B435-11E9-9278-68D0E5697425"}],"acknowledgement":"While working on this paper the authors were both supported by EPSRC grant EP/P026710/1, and the second author received additional support from the NWO Veni Grant 016.Veni.192.047. Thanks are due to Marta Pieropan, Arne Smeets and Sho Tanimoto for useful conversations related to this topic, and to the anonymous referee for numerous helpful suggestions.","ddc":["510"],"doi":"10.1007/s00209-021-02695-w","language":[{"iso":"eng"}],"page":"1071–1101","date_created":"2021-03-21T23:01:21Z","month":"03","isi":1,"publisher":"Springer Nature","status":"public","intvolume":"       299","publication":"Mathematische Zeitschrift","quality_controlled":"1","department":[{"_id":"TiBr"}],"oa_version":"Published Version","year":"2021","has_accepted_license":"1","article_type":"original","publication_identifier":{"eissn":["1432-1823"],"issn":["0025-5874"]},"external_id":{"isi":["000625573800002"]},"scopus_import":"1","date_updated":"2023-08-07T14:20:00Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No","_id":"9260","date_published":"2021-03-05T00:00:00Z","abstract":[{"text":"We study the density of rational points on a higher-dimensional orbifold (Pn−1,Δ) when Δ is a Q-divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.","lang":"eng"}],"file":[{"content_type":"application/pdf","file_id":"9279","relation":"main_file","success":1,"date_created":"2021-03-22T12:41:26Z","file_name":"2021_MathZeitschrift_Browning.pdf","creator":"dernst","file_size":492685,"date_updated":"2021-03-22T12:41:26Z","access_level":"open_access","checksum":"8ed9f49568806894744096dbbca0ad7b"}],"oa":1,"publication_status":"published","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"volume":299,"file_date_updated":"2021-03-22T12:41:26Z"}]