{"id":"https://openalex.org/W7213404659","doi":"https://doi.org/10.48550/arxiv.2609.16966","title":"Powers of the Thue--Morse Series: 2-Adic Valuations and Automatic Odd Parts","display_name":"Powers of the Thue--Morse Series: 2-Adic Valuations and Automatic Odd Parts","publication_year":2026,"publication_date":"2026-09-15","ids":{"openalex":"https://openalex.org/W7213404659","doi":"https://doi.org/10.48550/arxiv.2609.16966"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.16966","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16966","pdf_url":"https://arxiv.org/pdf/2609.16966","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/2609.16966","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5149321091","display_name":"Zhao Shen","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Shen, Zhao","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5149321091"],"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/T11428","display_name":"Advanced Mathematical Identities","score":0.8442000150680542,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.8442000150680542,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.11739999800920486,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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":0.020500000566244125,"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/thue-morse-sequence","display_name":"Thue-Morse sequence","score":0.8759999871253967},{"id":"https://openalex.org/keywords/2-adic-valuation","display_name":"2-adic valuation","score":0.9539999961853027},{"id":"https://openalex.org/keywords/automatic-sequences","display_name":"automatic sequences","score":0.8379999995231628},{"id":"https://openalex.org/keywords/binomial-coefficients","display_name":"binomial coefficients","score":0.8159999847412109},{"id":"https://openalex.org/keywords/combinatorics-on-words","display_name":"combinatorics on words","score":0.7070000171661377}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7192999720573425},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.6488999724388123},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.6279000043869019},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5415999889373779},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5324000120162964},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4966000020503998},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.46889999508857727},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.4618000090122223},{"id":"https://openalex.org/C186027771","wikidata":"https://www.wikidata.org/wiki/Q4008379","display_name":"Valuation (finance)","level":2,"score":0.44909998774528503},{"id":"https://openalex.org/C164680029","wikidata":"https://www.wikidata.org/wiki/Q1055293","display_name":"Multiple","level":2,"score":0.3644999861717224},{"id":"https://openalex.org/C33002781","wikidata":"https://www.wikidata.org/wiki/Q4826617","display_name":"Automaticity","level":3,"score":0.27790001034736633},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.263700008392334},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.26190000772476196}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.16966","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16966","pdf_url":"https://arxiv.org/pdf/2609.16966","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.2609.16966","doi":"https://doi.org/10.48550/arxiv.2609.16966","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.16966","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:2609.16966","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16966","pdf_url":"https://arxiv.org/pdf/2609.16966","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/W7213404659.pdf","grobid_xml":"https://content.openalex.org/works/W7213404659.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7166739924","https://openalex.org/W2092254834","https://openalex.org/W4288091111","https://openalex.org/W4384258982","https://openalex.org/W34795925","https://openalex.org/W2805118908","https://openalex.org/W4318144379","https://openalex.org/W4388974346","https://openalex.org/W2165820698","https://openalex.org/W2949093434"],"abstract_inverted_index":{"Let":[0],"$T(x)=\\prod_{j\\ge":[1],"0}(1-x^{2^j})=\\sum_{n\\ge":[2],"0}t(n)x^n$":[3],"be":[4,25],"the":[5,26,42,76],"Thue--Morse":[6],"generating":[7],"function,":[8],"and":[9,32,73,86],"write":[10],"$T(x)^m=\\sum_{n\\ge":[11],"0}t_m(n)x^n$":[12],"for":[13,65],"a":[14,19],"positive":[15],"integer":[16,21],"$m$.":[17],"For":[18,35,79],"nonzero":[20],"$a$,":[22],"let":[23],"$\u03bd_2(a)$":[24],"exponent":[27],"of":[28],"$2$":[29],"in":[30],"$a$":[31],"put":[33],"$\\operatorname{odd}(a)=a/2^{\u03bd_2(a)}$.":[34],"every":[36,93],"$s\\ge":[37],"1$,":[38,52],"we":[39,81],"prove":[40,82],"that":[41,88],"sequence":[43],"$(\\operatorname{odd}(t_m(n))\\bmod":[44],"2^s)_{n\\ge":[45],"0}$":[46],"is":[47,67],"$2$-automatic":[48],"when":[49],"$m=2^r$,":[50],"$r\\ge":[51,56],"or":[53],"$m=3\\cdot":[54],"2^r$,":[55],"2$.":[57],"In":[58],"both":[59],"families,":[60],"$\u03bd_2(t_m(n))=\u03bd_2\\binom{n+m-1}{m-1}$.":[61],"The":[62],"valuation":[63],"identity":[64],"$m=2^r$":[66],"known;":[68],"our":[69],"proof":[70],"recovers":[71],"it":[72],"also":[74],"establishes":[75],"automaticity":[77],"assertion.":[78],"$m=6$":[80],"$\u03bd_2(t_6(n))=\u03bd_2\\binom{n+5}{5}+\\mathbf{1}_{n\\equiv":[83],"3\\pmod":[84],"4}$":[85],"show":[87],"its":[89],"odd":[90],"parts":[91],"modulo":[92],"$2^s$":[94],"are":[95],"$2$-automatic.":[96]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-17T00:00:00"}
