{"id":"https://openalex.org/W7214031884","doi":"https://doi.org/10.48550/arxiv.2609.24752","title":"Density of almost squares of horospherical orbits in non-uniform quotients of $\\operatorname{SL}_2(\\mathbb{R})\\times\\operatorname{SL}_2(\\mathbb{R})$","display_name":"Density of almost squares of horospherical orbits in non-uniform quotients of $\\operatorname{SL}_2(\\mathbb{R})\\times\\operatorname{SL}_2(\\mathbb{R})$","publication_year":2026,"publication_date":"2026-09-21","ids":{"openalex":"https://openalex.org/W7214031884","doi":"https://doi.org/10.48550/arxiv.2609.24752"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.24752","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24752","pdf_url":"https://arxiv.org/pdf/2609.24752","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.24752","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5031020401","display_name":"Konstantin Andritsch","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Andritsch, Konstantin","raw_affiliation_strings":["Department of Mathematics , ETH Z\u00fcrich , Z\u00fcrich , Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , ETH Z\u00fcrich , Z\u00fcrich , Switzerland","institution_ids":["https://openalex.org/I35440088"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5031020401"],"corresponding_institution_ids":["https://openalex.org/I35440088"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.926800012588501,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.926800012588501,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":0.0406000018119812,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.004900000058114529,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/non-uniform-lattices","display_name":"non-uniform lattices","score":0.871999979019165},{"id":"https://openalex.org/keywords/homogeneous-dynamics","display_name":"homogeneous dynamics","score":0.7450000047683716},{"id":"https://openalex.org/keywords/shadowing-effect","display_name":"shadowing","score":0.5410000085830688},{"id":"https://openalex.org/keywords/periodic-orbits","display_name":"periodic orbits","score":0.7630000114440918},{"id":"https://openalex.org/keywords/weyl-s-inequality","display_name":"Weyl's inequality","score":0.8320000171661377},{"id":"https://openalex.org/keywords/recurrence","display_name":"recurrence","score":0.3580000102519989},{"id":"https://openalex.org/keywords/effective-equidistribution","display_name":"effective equidistribution","score":0.8339999914169312},{"id":"https://openalex.org/keywords/2-torus","display_name":"2-torus","score":0.6549999713897705}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7383000254631042},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.664900004863739},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.6189000010490417},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.5799999833106995},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.46790000796318054},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4422999918460846},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.42910000681877136},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.3686999976634979},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36570000648498535},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3612000048160553},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.35519999265670776},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.3077999949455261},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.28369998931884766},{"id":"https://openalex.org/C527412718","wikidata":"https://www.wikidata.org/wiki/Q855395","display_name":"Interpretation (philosophy)","level":2,"score":0.28110000491142273},{"id":"https://openalex.org/C87945829","wikidata":"https://www.wikidata.org/wiki/Q1353233","display_name":"Conjugacy class","level":2,"score":0.26089999079704285},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.25519999861717224}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.24752","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24752","pdf_url":"https://arxiv.org/pdf/2609.24752","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.24752","doi":"https://doi.org/10.48550/arxiv.2609.24752","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.24752","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.24752","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24752","pdf_url":"https://arxiv.org/pdf/2609.24752","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1168414883","display_name":"Equidistribution in Number Theory","funder_award_id":"10003145","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320309370","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10"},{"id":"https://openalex.org/F4320320924","display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","ror":"https://ror.org/00yjd3n13"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214031884.pdf","grobid_xml":"https://content.openalex.org/works/W7214031884.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4392078172","https://openalex.org/W7101747524","https://openalex.org/W2535829653","https://openalex.org/W2964348108","https://openalex.org/W2891877792","https://openalex.org/W2285395216","https://openalex.org/W3100823020","https://openalex.org/W2066391028","https://openalex.org/W2616889728","https://openalex.org/W2949098958"],"abstract_inverted_index":{"Let":[0,14],"$\u0393\\subset\\operatorname{SL}_2(\\mathbb{R})\\times\\operatorname{SL}_2(\\mathbb{R})$":[1],"be":[2,16],"an":[3,82],"irreducible,":[4],"non-uniform":[5,145],"lattice":[6],"and":[7,29,59,116],"define":[8],"the":[9,17,35,51,61,68,75,90,113,126,129,141],"space":[10],"$X":[11],"=":[12],"\\operatorname{SL}_2(\\mathbb{R})\\times\\operatorname{SL}_2(\\mathbb{R})/\u0393$.":[13],"$U$":[15],"standard":[18],"horospherical":[19,109,121],"subgroup":[20],"in":[21,45,147],"$\\operatorname{SL}_2(\\mathbb{R})\\times\\operatorname{SL}_2(\\mathbb{R})$.":[22],"We":[23],"show":[24],"that":[25],"for":[26,74,144],"every":[27],"$\u03b4>0$":[28],"$x\\in":[30],"X$":[31],"with":[32],"dense":[33,44],"$U$-orbit,":[34,70],"$U$-orbit":[36],"evaluated":[37],"at":[38],"almost":[39,138],"squares,":[40],"$\\{(u_{(n^{2-\u03b4},m^{2-\u03b4})}~:~n,m\\in\\mathbb{N}\\}\\cdot":[41],"x$,":[42],"is":[43,53,98],"$X$.":[46],"The":[47,92],"main":[48],"idea":[49],"of":[50,85,95,108,119,128,132,137,140],"proof":[52],"to":[54,63,88],"shadow":[55],"large":[56],"periodic":[57,69,96],"$U$-orbits":[58,97],"reduce":[60],"statement":[62,66,73],"a":[64,72],"density":[65,136],"within":[67],"i.e.":[71],"$2$-torus":[76],"$\\mathbb{T}^2$.":[77],"This":[78],"then":[79],"follows":[80,125],"from":[81,102],"effective":[83,93,117],"version":[84],"Weyl's":[86],"inequality":[87],"deduce":[89],"result.":[91],"shadowing":[94],"achieved":[99],"using":[100],"tools":[101],"homogeneous":[103],"dynamics,":[104],"namely":[105],"quantitative":[106],"non-divergence":[107],"subgroups,":[110],"recurrence":[111],"under":[112],"diagonal":[114],"flow":[115,143],"equidistribution":[118],"expanding":[120],"subgroups.":[122],"Our":[123],"approach":[124],"strategy":[127],"work":[130],"\\cite{KR25}":[131],"\\citeauthor{KR25}":[133],"who":[134],"established":[135],"squares":[139],"horocycle":[142],"lattices":[146],"$\\operatorname{SL}_2(\\mathbb{R})$.":[148]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-23T00:00:00"}
