[{"quality_controlled":"1","author":[{"last_name":"Elsholtz","first_name":"Christian","full_name":"Elsholtz, Christian"},{"first_name":"Lena","full_name":"Wurzinger, Lena","orcid":"0009-0004-5360-0074","last_name":"Wurzinger","id":"50c57d72-32a8-11ee-aeea-d652094d2ccd"}],"PlanS_conform":"1","publisher":"Oxford University Press","ddc":["510"],"type":"journal_article","publication_status":"published","file":[{"date_updated":"2025-01-28T07:03:51Z","access_level":"open_access","checksum":"1a06e052761d3f1e873463d6f529dd82","file_size":424645,"success":1,"date_created":"2025-01-28T07:03:51Z","file_id":"18931","relation":"main_file","creator":"dernst","content_type":"application/pdf","file_name":"2024_QuarterlyJourMath_Elsholtz.pdf"}],"title":"Sumsets in the set of squares","department":[{"_id":"TiBr"}],"article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","issue":"4","date_updated":"2025-12-04T14:46:28Z","volume":75,"year":"2024","day":"01","isi":1,"acknowledgement":"This manuscript grew out of the second author’s MSc Thesis at Graz University of Technology [34]. C. Elsholtz is supported by a joint FWF-ANR project ArithRand, grant numbers FWF I 4945-N and ANR-20-CE91-0006. Both authors would like to thank Igor Shparlinski for drawing our attention to related character sum estimates. Furthermore, we would like to thank the referee for a careful reading of the paper.","file_date_updated":"2025-01-28T07:03:51Z","article_type":"original","external_id":{"isi":["001304396600001"]},"OA_place":"publisher","abstract":[{"lang":"eng","text":"We study sumsets 𝒜 + ℬ in the set of squares 𝒮 (and, more generally, in the set of kth powers 𝒮k, where k ≥2 is an integer). It is known by a result of Gyarmati that 𝒜 + ℬ ⊂ 𝒮k ∩[1,N] implies that min(|𝒜|,|ℬ|) =Ok(logN). Here, we study how the upper bound on |ℬ| decreases, when the size of |𝒜| increases (or vice versa). In particular, if |𝒜| ≥ Ck1m m(logN)1m , then |ℬ| = Ok(m2logN), for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with m ∼ loglogN this gives: If |𝒜| ≥ CkloglogN,then |ℬ| = Ok(logN(loglogN)2)."}],"scopus_import":"1","publication":"The Quarterly Journal of Mathematics","month":"12","date_created":"2025-01-28T06:55:31Z","oa":1,"OA_type":"hybrid","intvolume":"        75","_id":"18930","corr_author":"1","citation":{"short":"C. Elsholtz, L. Wurzinger, The Quarterly Journal of Mathematics 75 (2024) 1243–1254.","mla":"Elsholtz, Christian, and Lena Wurzinger. “Sumsets in the Set of Squares.” <i>The Quarterly Journal of Mathematics</i>, vol. 75, no. 4, Oxford University Press, 2024, pp. 1243–54, doi:<a href=\"https://doi.org/10.1093/qmath/haae044\">10.1093/qmath/haae044</a>.","ista":"Elsholtz C, Wurzinger L. 2024. Sumsets in the set of squares. The Quarterly Journal of Mathematics. 75(4), 1243–1254.","ama":"Elsholtz C, Wurzinger L. Sumsets in the set of squares. <i>The Quarterly Journal of Mathematics</i>. 2024;75(4):1243-1254. doi:<a href=\"https://doi.org/10.1093/qmath/haae044\">10.1093/qmath/haae044</a>","apa":"Elsholtz, C., &#38; Wurzinger, L. (2024). Sumsets in the set of squares. <i>The Quarterly Journal of Mathematics</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/qmath/haae044\">https://doi.org/10.1093/qmath/haae044</a>","ieee":"C. Elsholtz and L. Wurzinger, “Sumsets in the set of squares,” <i>The Quarterly Journal of Mathematics</i>, vol. 75, no. 4. Oxford University Press, pp. 1243–1254, 2024.","chicago":"Elsholtz, Christian, and Lena Wurzinger. “Sumsets in the Set of Squares.” <i>The Quarterly Journal of Mathematics</i>. Oxford University Press, 2024. <a href=\"https://doi.org/10.1093/qmath/haae044\">https://doi.org/10.1093/qmath/haae044</a>."},"oa_version":"Published Version","doi":"10.1093/qmath/haae044","page":"1243-1254","language":[{"iso":"eng"}],"publication_identifier":{"issn":["0033-5606"],"eissn":["1464-3847"]},"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)"},"date_published":"2024-12-01T00:00:00Z","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"abstract":[{"text":"As a discrete analogue of Kac’s celebrated question on ‘hearing the shape of a drum’ and towards a practical\r\ngraph isomorphism test, it is of interest to understand which graphs are determined up to isomorphism by\r\ntheir spectrum (of their adjacency matrix). A striking conjecture in this area, due to van Dam and Haemers,\r\nis that ‘almost all graphs are determined by their spectrum’, meaning that the fraction of unlabelled n-vertex\r\ngraphs which are determined by their spectrum converges to 1 as n → ∞.\r\nIn this paper, we make a step towards this conjecture, showing that there are exponentially many n-vertex\r\ngraphs which are determined by their spectrum. This improves on previous bounds (of shape e\r\nc\r\n√\r\nn\r\n). We also\r\npropose a number of further directions of research.\r\n","lang":"eng"}],"scopus_import":"1","month":"06","publication":"Quarterly Journal of Mathematics","date_created":"2024-09-01T22:01:07Z","oa":1,"corr_author":"1","_id":"17475","intvolume":"        75","doi":"10.1093/qmath/haae030","oa_version":"Published Version","citation":{"ama":"Koval I, Kwan MA. Exponentially many graphs are determined by their spectrum. <i>Quarterly Journal of Mathematics</i>. 2024;75(3):869-899. doi:<a href=\"https://doi.org/10.1093/qmath/haae030\">10.1093/qmath/haae030</a>","ieee":"I. Koval and M. A. Kwan, “Exponentially many graphs are determined by their spectrum,” <i>Quarterly Journal of Mathematics</i>, vol. 75, no. 3. Oxford University Press, pp. 869–899, 2024.","apa":"Koval, I., &#38; Kwan, M. A. (2024). Exponentially many graphs are determined by their spectrum. <i>Quarterly Journal of Mathematics</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/qmath/haae030\">https://doi.org/10.1093/qmath/haae030</a>","chicago":"Koval, Illya, and Matthew Alan Kwan. “Exponentially Many Graphs Are Determined by Their Spectrum.” <i>Quarterly Journal of Mathematics</i>. Oxford University Press, 2024. <a href=\"https://doi.org/10.1093/qmath/haae030\">https://doi.org/10.1093/qmath/haae030</a>.","short":"I. Koval, M.A. Kwan, Quarterly Journal of Mathematics 75 (2024) 869–899.","ista":"Koval I, Kwan MA. 2024. Exponentially many graphs are determined by their spectrum. Quarterly Journal of Mathematics. 75(3), 869–899.","mla":"Koval, Illya, and Matthew Alan Kwan. “Exponentially Many Graphs Are Determined by Their Spectrum.” <i>Quarterly Journal of Mathematics</i>, vol. 75, no. 3, Oxford University Press, 2024, pp. 869–99, doi:<a href=\"https://doi.org/10.1093/qmath/haae030\">10.1093/qmath/haae030</a>."},"page":"869-899","language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1464-3847"],"issn":["0033-5606"]},"arxiv":1,"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)"},"date_published":"2024-06-19T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","status":"public","author":[{"id":"2eed1f3b-896a-11ed-bdf8-93c7c4bf159e","last_name":"Koval","full_name":"Koval, Illya","first_name":"Illya"},{"full_name":"Kwan, Matthew Alan","first_name":"Matthew Alan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","orcid":"0000-0002-4003-7567","last_name":"Kwan"}],"quality_controlled":"1","ddc":["500"],"publisher":"Oxford University Press","publication_status":"published","type":"journal_article","file":[{"file_id":"17851","date_created":"2024-09-06T12:23:57Z","success":1,"file_size":946411,"file_name":"2024_QuJofMath_Koval.pdf","creator":"cchlebak","content_type":"application/pdf","relation":"main_file","date_updated":"2024-09-06T12:23:57Z","checksum":"abf200d37ad69e6f2c0750a30296ad97","access_level":"open_access"}],"article_processing_charge":"Yes (via OA deal)","title":"Exponentially many graphs are determined by their spectrum","department":[{"_id":"MaKw"},{"_id":"VaKa"}],"date_updated":"2025-09-08T09:09:41Z","issue":"3","has_accepted_license":"1","year":"2024","volume":75,"day":"19","isi":1,"acknowledgement":"Matthew Kwan was supported by ERC Starting Grant ‘RANDSTRUCT’ No. 101076777.","file_date_updated":"2024-09-06T12:23:57Z","article_type":"original","external_id":{"isi":["001249741500001"],"arxiv":["2309.09788"]},"project":[{"grant_number":"101076777","name":"Randomness and structure in combinatorics","_id":"bd95085b-d553-11ed-ba76-e55d3349be45"}]},{"publisher":"Oxford University Press","ddc":["510"],"author":[{"last_name":"Horesh","id":"C8B7BF48-8D81-11E9-BCA9-F536E6697425","first_name":"Tal","full_name":"Horesh, Tal"},{"last_name":"Karasik","first_name":"Yakov","full_name":"Karasik, Yakov"}],"quality_controlled":"1","article_processing_charge":"Yes (via OA deal)","department":[{"_id":"TiBr"}],"title":"Equidistribution of primitive lattices in ℝn","date_updated":"2025-09-09T14:03:32Z","issue":"4","has_accepted_license":"1","publication_status":"published","type":"journal_article","file":[{"checksum":"bf29baa9eae8500f3374dbcb80712687","access_level":"open_access","date_updated":"2024-01-02T07:37:09Z","file_name":"2023_QuarterlyJourMath_Horesh.pdf","content_type":"application/pdf","creator":"dernst","relation":"main_file","file_id":"14720","date_created":"2024-01-02T07:37:09Z","file_size":724748,"success":1}],"isi":1,"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.","year":"2023","volume":74,"day":"01","article_type":"original","external_id":{"isi":["001005945400001"],"arxiv":["2012.04508"]},"project":[{"_id":"26A8D266-B435-11E9-9278-68D0E5697425","name":"Between rational and integral points","grant_number":"EP-P026710-2"}],"file_date_updated":"2024-01-02T07:37:09Z","date_created":"2023-12-31T23:01:03Z","month":"12","publication":"Quarterly Journal of Mathematics","oa":1,"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."}],"scopus_import":"1","page":"1253-1294","language":[{"iso":"eng"}],"corr_author":"1","_id":"14717","intvolume":"        74","oa_version":"Published Version","doi":"10.1093/qmath/haad008","citation":{"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>.","ista":"Horesh T, Karasik Y. 2023. Equidistribution of primitive lattices in ℝn. Quarterly Journal of Mathematics. 74(4), 1253–1294.","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>","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.","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>.","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>"},"publication_identifier":{"issn":["0033-5606"],"eissn":["1464-3847"]},"arxiv":1,"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)"},"date_published":"2023-12-01T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","status":"public"}]