{"id":"https://openalex.org/W7223936743","doi":"https://doi.org/10.48550/arxiv.2610.34288","title":"Cross numbers and the characterization of finite abelian groups by length sets","display_name":"Cross numbers and the characterization of finite abelian groups by length sets","publication_year":2026,"publication_date":"2026-10-25","ids":{"openalex":"https://openalex.org/W7223936743","doi":"https://doi.org/10.48550/arxiv.2610.34288"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.34288","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.34288","pdf_url":"https://arxiv.org/pdf/2610.34288","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/2610.34288","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029815260","display_name":"Qinghai Zhong","orcid":"https://orcid.org/0000-0003-2096-0810","observed_orcids":["https://orcid.org/0000-0003-2096-0810"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zhong, Qinghai","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5029815260"],"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.4934000074863434,"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.4934000074863434,"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":0.13809999823570251,"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":0.09149999916553497,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/finite-abelian-groups","display_name":"finite abelian groups","score":0.8539999723434448},{"id":"https://openalex.org/keywords/characterization-problem","display_name":"characterization problem","score":0.7300000190734863},{"id":"https://openalex.org/keywords/davenport-constant","display_name":"Davenport constant","score":0.9580000042915344},{"id":"https://openalex.org/keywords/non-unique-factorization","display_name":"non-unique factorization","score":0.8420000076293945},{"id":"https://openalex.org/keywords/zero-sum-sequences","display_name":"zero-sum sequences","score":0.8360000252723694},{"id":"https://openalex.org/keywords/ideal-class-group","display_name":"ideal class group","score":0.8500000238418579},{"id":"https://openalex.org/keywords/inverse-problems","display_name":"inverse problems","score":0.4300000071525574}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8835999965667725},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.838699996471405},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7117999792098999},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.59579998254776},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5741999745368958},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.5514000058174133},{"id":"https://openalex.org/C81008192","wikidata":"https://www.wikidata.org/wiki/Q1017231","display_name":"Elementary abelian group","level":3,"score":0.5146999955177307},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5062999725341797},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.4918000102043152},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.453000009059906},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.43860000371932983},{"id":"https://openalex.org/C197623797","wikidata":"https://www.wikidata.org/wiki/Q1454111","display_name":"Free abelian group","level":5,"score":0.426800012588501},{"id":"https://openalex.org/C20725272","wikidata":"https://www.wikidata.org/wiki/Q245462","display_name":"Cyclic group","level":3,"score":0.41940000653266907},{"id":"https://openalex.org/C200681373","wikidata":"https://www.wikidata.org/wiki/Q1260803","display_name":"Rank of an abelian group","level":4,"score":0.4117000102996826},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4050999879837036},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.38670000433921814},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.38580000400543213},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.36480000615119934},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.3427000045776367},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.34220001101493835},{"id":"https://openalex.org/C2779483109","wikidata":"https://www.wikidata.org/wiki/Q4681347","display_name":"Additive group","level":3,"score":0.3285999894142151},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.3125},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.2994999885559082},{"id":"https://openalex.org/C2777404646","wikidata":"https://www.wikidata.org/wiki/Q1057968","display_name":"Finite group","level":3,"score":0.29760000109672546},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.29739999771118164},{"id":"https://openalex.org/C186746360","wikidata":"https://www.wikidata.org/wiki/Q759832","display_name":"Solvable group","level":3,"score":0.2612999975681305},{"id":"https://openalex.org/C8367249","wikidata":"https://www.wikidata.org/wiki/Q912083","display_name":"Ideal class group","level":3,"score":0.2583000063896179}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.34288","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.34288","pdf_url":"https://arxiv.org/pdf/2610.34288","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.2610.34288","doi":"https://doi.org/10.48550/arxiv.2610.34288","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.34288","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:2610.34288","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.34288","pdf_url":"https://arxiv.org/pdf/2610.34288","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/W7223936743.pdf","grobid_xml":"https://content.openalex.org/works/W7223936743.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2313382455","https://openalex.org/W2075209367","https://openalex.org/W2962872531","https://openalex.org/W4392781796","https://openalex.org/W3105535124","https://openalex.org/W2338396821","https://openalex.org/W4205425787","https://openalex.org/W2167563230","https://openalex.org/W2041111052","https://openalex.org/W2991930528"],"abstract_inverted_index":{"The":[0,105],"Characterization":[1],"Problem":[2],"asks":[3,37],"whether":[4,38],"a":[5,28,91,109,144],"finite":[6,92],"abelian":[7,93],"group":[8,70,94,176],"with":[9,95,132,183],"Davenport":[10],"constant":[11],"at":[12],"least":[13],"four":[14],"is":[15,27,90,111,143,168],"determined,":[16],"up":[17],"to":[18,48],"isomorphism,":[19],"by":[20,170],"its":[21,121,136],"system":[22],"of":[23,42,56,58,108,114,117,120,126,159,173],"length":[24,128],"sets.":[25],"This":[26,123],"central":[29],"inverse":[30],"problem":[31,82],"in":[32],"non-unique":[33],"factorization":[34,76],"theory:":[35],"it":[36],"the":[39,50,67,112,115,118,155,171,174],"possible":[40],"numbers":[41],"irreducible":[43],"factors":[44],"retain":[45],"enough":[46],"information":[47],"recover":[49],"underlying":[51],"class":[52,69],"group.":[53],"For":[54,179],"rings":[55],"integers":[57],"algebraic":[59],"number":[60,107,158],"fields,":[61],"an":[62,151],"affirmative":[63],"answer":[64],"means":[65],"that":[66],"ideal":[68],"can":[71],"be":[72],"recovered":[73],"from":[74],"these":[75],"lengths":[77],"alone.":[78],"We":[79,148],"resolve":[80],"this":[81],"for":[83,139,154],"all":[84],"homocyclic":[85],"groups":[86],"$C_n^r$:":[87],"if":[88],"$G$":[89],"$\\mathsf":[96],"D(G)\\ge4$":[97],"and":[98,134],"$\\mathcal":[99],"L(G)=\\mathcal":[100],"L(C_n^r)$,":[101],"then":[102],"$G\\cong":[103,180],"C_n^r$.":[104],"cross":[106,157],"sequence":[110,127],"sum":[113],"reciprocals":[116],"orders":[119],"terms.":[122],"weighted":[124],"measure":[125],"links":[129],"zero-sum":[130,141,161],"theory":[131],"half-factoriality,":[133],"determining":[135],"maximal":[137,156],"value":[138],"minimal":[140,160],"sequences":[142,162],"longstanding":[145],"extremal":[146],"problem.":[147],"also":[149],"obtain":[150],"exact":[152],"formula":[153],"over":[163],"subsets":[164],"whose":[165],"minimum":[166],"distance":[167],"divisible":[169],"rank":[172],"ambient":[175],"minus":[177],"one.":[178],"C_{n_1}\\oplus\\cdots\\oplus":[181],"C_{n_r}$":[182],"$1":[184]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-28T00:00:00"}