{"id":"https://openalex.org/W7212005728","doi":"https://doi.org/10.48550/arxiv.2609.07714","title":"The spectrum of $(\u03be\u03b1^n)$ can be uncountable","display_name":"The spectrum of $(\u03be\u03b1^n)$ can be uncountable","publication_year":2026,"publication_date":"2026-09-07","ids":{"openalex":"https://openalex.org/W7212005728","doi":"https://doi.org/10.48550/arxiv.2609.07714"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.07714","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.07714","pdf_url":"https://arxiv.org/pdf/2609.07714","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/2609.07714","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102947803","display_name":"Hikmet Burak \u00d6zcan","orcid":"https://orcid.org/0000-0002-8684-7311","observed_orcids":["https://orcid.org/0000-0002-8684-7311"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"\u00d6zcan, Hikmet Burak","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5102947803"],"corresponding_institution_ids":[],"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":0.47119998931884766,"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":0.47119998931884766,"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/T11166","display_name":"Analytic Number Theory Research","score":0.45559999346733093,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.006899999920278788,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/uniform-distribution-modulo-1","display_name":"uniform distribution modulo one","score":0.7770000100135803},{"id":"https://openalex.org/keywords/diophantine-approximation","display_name":"Diophantine approximation","score":0.7200000286102295},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.6439999938011169},{"id":"https://openalex.org/keywords/exponential-sequences","display_name":"exponential sequences","score":0.5490000247955322},{"id":"https://openalex.org/keywords/distribution-modulo-one","display_name":"distribution modulo one","score":0.4909999966621399}],"concepts":[{"id":"https://openalex.org/C142399903","wikidata":"https://www.wikidata.org/wiki/Q1128796","display_name":"Uncountable set","level":3,"score":0.9193999767303467},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.7935000061988831},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.73089998960495},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.713699996471405},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7009999752044678},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.6486999988555908},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.574999988079071},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49219998717308044},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48410001397132874},{"id":"https://openalex.org/C40422974","wikidata":"https://www.wikidata.org/wiki/Q741248","display_name":"Assertion","level":2,"score":0.4713999927043915},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4699000120162964},{"id":"https://openalex.org/C102966492","wikidata":"https://www.wikidata.org/wiki/Q1227061","display_name":"Diophantine approximation","level":3,"score":0.45089998841285706},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.41290000081062317},{"id":"https://openalex.org/C94931360","wikidata":"https://www.wikidata.org/wiki/Q607728","display_name":"Irrational number","level":2,"score":0.39640000462532043},{"id":"https://openalex.org/C2780297109","wikidata":"https://www.wikidata.org/wiki/Q917390","display_name":"Essential spectrum","level":3,"score":0.34450000524520874},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.3359000086784363},{"id":"https://openalex.org/C206530611","wikidata":"https://www.wikidata.org/wiki/Q905896","display_name":"Diophantine equation","level":2,"score":0.3319999873638153},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.32409998774528503},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.30630001425743103},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.30320000648498535},{"id":"https://openalex.org/C169611556","wikidata":"https://www.wikidata.org/wiki/Q230689","display_name":"Contractible space","level":2,"score":0.29600000381469727},{"id":"https://openalex.org/C36605576","wikidata":"https://www.wikidata.org/wiki/Q273188","display_name":"Cantor set","level":2,"score":0.28540000319480896},{"id":"https://openalex.org/C55439883","wikidata":"https://www.wikidata.org/wiki/Q360812","display_name":"Correctness","level":2,"score":0.28540000319480896},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28459998965263367},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.2784999907016754},{"id":"https://openalex.org/C61067352","wikidata":"https://www.wikidata.org/wiki/Q1897429","display_name":"Ultrametric space","level":3,"score":0.2721000015735626},{"id":"https://openalex.org/C2778408900","wikidata":"https://www.wikidata.org/wiki/Q1329258","display_name":"Cardinal number (linguistics)","level":2,"score":0.26429998874664307},{"id":"https://openalex.org/C199879729","wikidata":"https://www.wikidata.org/wiki/Q205140","display_name":"Infinite set","level":3,"score":0.25769999623298645},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2572000026702881},{"id":"https://openalex.org/C80845027","wikidata":"https://www.wikidata.org/wiki/Q15144062","display_name":"Continuous spectrum","level":2,"score":0.2558000087738037},{"id":"https://openalex.org/C55375708","wikidata":"https://www.wikidata.org/wiki/Q1201815","display_name":"Null set","level":3,"score":0.25119999051094055}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.07714","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.07714","pdf_url":"https://arxiv.org/pdf/2609.07714","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.2609.07714","doi":"https://doi.org/10.48550/arxiv.2609.07714","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.07714","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.07714","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.07714","pdf_url":"https://arxiv.org/pdf/2609.07714","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212005728.pdf","grobid_xml":"https://content.openalex.org/works/W7212005728.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7154569998"],"related_works":["https://openalex.org/W7098554536","https://openalex.org/W1981999557","https://openalex.org/W4361807111","https://openalex.org/W2272270943","https://openalex.org/W2318138550","https://openalex.org/W825225741","https://openalex.org/W2070558513","https://openalex.org/W2906691739","https://openalex.org/W4408556912","https://openalex.org/W2022695828"],"abstract_inverted_index":{"In":[0],"this":[1],"note,":[2],"we":[3,66],"give":[4],"a":[5],"counterexample":[6],"to":[7,25],"the":[8,32,35,40,81,88],"assertion":[9],"of":[10,34,42,83],"Problem":[11],"10.4":[12],"in":[13],"Bugeaud's":[14],"monograph":[15],"Distribution":[16],"modulo":[17,52],"one":[18,86],"and":[19,62,87],"Diophantine":[20],"approximation,":[21],"which":[22,46,80],"goes":[23],"back":[24],"Mend\u00e8s":[26],"France.":[27],"The":[28],"problem":[29],"states":[30],"that":[31,38,68],"spectrum":[33,82],"sequence":[36],"$(\u03be\u03b1^n)_{n\\ge1}$,":[37],"is,":[39],"set":[41],"irrational":[43],"$\u03b8\\in(0,1)$":[44],"for":[45,58,69,79],"$(\u03be\u03b1^n-n\u03b8)_{n\\ge1}$":[47],"is":[48,54],"not":[49],"uniformly":[50],"distributed":[51],"one,":[53],"at":[55],"most":[56],"countable":[57],"all":[59],"real":[60,71,76],"$\u03be\\ne0$":[61],"$\u03b1>1$.":[63],"More":[64],"precisely,":[65],"prove":[67],"every":[70],"$\u03b1>1$":[72],"there":[73],"are":[74],"$2^{\\aleph_0}$":[75],"numbers":[77],"$\u03be>0$":[78],"$(\u03be\u03b1^n)_{n\\ge1}$":[84],"contains":[85],"same":[89],"uncountable":[90],"set.":[91]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-10T00:00:00"}