{"id":"https://openalex.org/W7220706998","doi":"https://doi.org/10.48550/arxiv.2610.03787","title":"Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum","display_name":"Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum","publication_year":2026,"publication_date":"2026-09-30","ids":{"openalex":"https://openalex.org/W7220706998","doi":"https://doi.org/10.48550/arxiv.2610.03787"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.03787","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.03787","pdf_url":"https://arxiv.org/pdf/2610.03787","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/2610.03787","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5114094830","display_name":"Maksim Lavrov","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I19880235","display_name":"Lomonosov Moscow State University","ror":"https://ror.org/010pmpe69","country_code":"RU","type":"education","lineage":["https://openalex.org/I19880235"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"Lavrov, Maksim","raw_affiliation_strings":["Faculty of Mechanics and Mathematics , Lomonosov Moscow State University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Mechanics and Mathematics , Lomonosov Moscow State University","institution_ids":["https://openalex.org/I19880235"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5114094830"],"corresponding_institution_ids":["https://openalex.org/I19880235"],"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/T11166","display_name":"Analytic Number Theory Research","score":0.7294999957084656,"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"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.7294999957084656,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.19099999964237213,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.0210999995470047,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/dirichlet-spectrum","display_name":"Dirichlet spectrum","score":0.9309999942779541},{"id":"https://openalex.org/keywords/isolated-points","display_name":"isolated points","score":0.9580000042915344},{"id":"https://openalex.org/keywords/continued-fractions","display_name":"continued fractions","score":0.6790000200271606},{"id":"https://openalex.org/keywords/computer-assisted-proof","display_name":"computer-assisted proof","score":0.7139999866485596},{"id":"https://openalex.org/keywords/irrational-numbers","display_name":"irrational numbers","score":0.6610000133514404}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7006000280380249},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6977999806404114},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.6255999803543091},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.5497000217437744},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.53329998254776},{"id":"https://openalex.org/C207803063","wikidata":"https://www.wikidata.org/wiki/Q620595","display_name":"Dirichlet series","level":4,"score":0.5127000212669373},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.37279999256134033},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3637999892234802},{"id":"https://openalex.org/C148158259","wikidata":"https://www.wikidata.org/wiki/Q975727","display_name":"Dirichlet's principle","level":4,"score":0.3504999876022339},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.34450000524520874},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3407000005245209},{"id":"https://openalex.org/C102231394","wikidata":"https://www.wikidata.org/wiki/Q5280763","display_name":"Dirichlet eigenvalue","level":5,"score":0.3312000036239624},{"id":"https://openalex.org/C94931360","wikidata":"https://www.wikidata.org/wiki/Q607728","display_name":"Irrational number","level":2,"score":0.3199999928474426},{"id":"https://openalex.org/C187967104","wikidata":"https://www.wikidata.org/wiki/Q2264419","display_name":"Dirichlet L-function","level":5,"score":0.31150001287460327},{"id":"https://openalex.org/C2779560616","wikidata":"https://www.wikidata.org/wiki/Q1192869","display_name":"Dirichlet problem","level":3,"score":0.2915000021457672},{"id":"https://openalex.org/C103302479","wikidata":"https://www.wikidata.org/wiki/Q1227685","display_name":"Dirichlet conditions","level":5,"score":0.2888999879360199},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.2833000123500824},{"id":"https://openalex.org/C96234433","wikidata":"https://www.wikidata.org/wiki/Q5280760","display_name":"Dirichlet's energy","level":4,"score":0.2797999978065491},{"id":"https://openalex.org/C20444455","wikidata":"https://www.wikidata.org/wiki/Q906024","display_name":"Dirichlet kernel","level":5,"score":0.26249998807907104},{"id":"https://openalex.org/C52290693","wikidata":"https://www.wikidata.org/wiki/Q5532435","display_name":"Generalized Dirichlet distribution","level":5,"score":0.2524999976158142}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.03787","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.03787","pdf_url":"https://arxiv.org/pdf/2610.03787","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.2610.03787","doi":"https://doi.org/10.48550/arxiv.2610.03787","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.03787","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:2610.03787","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.03787","pdf_url":"https://arxiv.org/pdf/2610.03787","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":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220706998.pdf","grobid_xml":"https://content.openalex.org/works/W7220706998.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W2032649907","https://openalex.org/W7161963953"],"related_works":["https://openalex.org/W2032649907","https://openalex.org/W7161963953","https://openalex.org/W4407425759","https://openalex.org/W4412651050","https://openalex.org/W3215856570","https://openalex.org/W2213052011","https://openalex.org/W1576957987","https://openalex.org/W7164008935","https://openalex.org/W2006785250","https://openalex.org/W6922915924"],"abstract_inverted_index":{"We":[0,36,73],"study":[1],"the":[2,11,15,19,28,44,49,60,89,95,99,106],"one-dimensional":[3],"Dirichlet":[4,45,90,114],"spectrum":[5,20],"and":[6,21,104],"obtain":[7],"new":[8],"results":[9],"concerning":[10],"origin":[12],"$\u03bc$":[13],"of":[14,59,88,98,109],"ray":[16],"contained":[17],"in":[18,43],"its":[22],"isolated":[23,86],"points.":[24],"First,":[25],"we":[26,93],"prove":[27,75],"improved":[29,50],"upper":[30],"bound":[31,52,62],"$\u03bc\\leq":[32],"(448875+1611\\sqrt{67773})/46982":[33],"=":[34,55],"18.4809\\ldots$.":[35],"also":[37],"exhibit":[38],"an":[39,64],"explicit":[40],"gap":[41],"$(L,R)$":[42],"spectrum,":[46],"which":[47],"yields":[48],"lower":[51,61],"$\u03bc\\geq":[53],"R":[54],"5.9596\\ldots$.":[56],"The":[57],"proof":[58],"includes":[63],"exact":[65],"computer-assisted":[66],"finite":[67],"search":[68],"over":[69],"admissible":[70],"continued-fraction":[71],"words.":[72],"further":[74],"that":[76],"all":[77,110],"points":[78,87],"$Z_k$,":[79],"$k\\geq":[80],"2$,":[81],"introduced":[82],"by":[83],"Ivanov":[84],"are":[85],"spectrum.":[91],"Finally,":[92],"determine":[94],"asymptotic":[96],"behavior":[97],"neighboring":[100],"gaps":[101],"around":[102],"$Z_k$":[103],"describe":[105],"equivalence":[107],"classes":[108],"irrational":[111],"numbers":[112],"whose":[113],"value":[115],"is":[116],"$Z_k$.":[117]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}