{"id":"https://openalex.org/W7212123187","doi":"https://doi.org/10.48550/arxiv.2609.10148","title":"Infinite Families of Counterexamples to a Conjecture of Liu and Qian and a Refined Inverse Theorem for Restricted Sumsets in $\\mathbb{Z}_p$","display_name":"Infinite Families of Counterexamples to a Conjecture of Liu and Qian and a Refined Inverse Theorem for Restricted Sumsets in $\\mathbb{Z}_p$","publication_year":2026,"publication_date":"2026-09-09","ids":{"openalex":"https://openalex.org/W7212123187","doi":"https://doi.org/10.48550/arxiv.2609.10148"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.10148","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10148","pdf_url":"https://arxiv.org/pdf/2609.10148","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/2609.10148","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100664432","display_name":"Jiantao Li","orcid":"https://orcid.org/0009-0002-6402-3773","observed_orcids":["https://orcid.org/0009-0002-6402-3773"]},"institutions":[{"id":"https://openalex.org/I118803816","display_name":"Liaoning University","ror":"https://ror.org/03xpwj629","country_code":"CN","type":"education","lineage":["https://openalex.org/I118803816"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Li, Jiantao","raw_affiliation_strings":["School of Mathematics and Statistics , Liaoning University , Shenyang , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics , Liaoning University , Shenyang , China","institution_ids":["https://openalex.org/I118803816"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5125229510","display_name":"X. Liang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I118803816","display_name":"Liaoning University","ror":"https://ror.org/03xpwj629","country_code":"CN","type":"education","lineage":["https://openalex.org/I118803816"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Liang, Xinyi","raw_affiliation_strings":["School of Mathematics and Statistics , Liaoning University , Shenyang , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics , Liaoning University , Shenyang , China","institution_ids":["https://openalex.org/I118803816"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I118803816"],"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.7813000082969666,"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"}},"topics":[{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.7813000082969666,"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":0.1096000000834465,"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":0.08100000023841858,"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/restricted-sumsets","display_name":"restricted sumsets","score":0.9589999914169312},{"id":"https://openalex.org/keywords/critical-pairs","display_name":"critical pairs","score":0.7170000076293945},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.7799999713897705},{"id":"https://openalex.org/keywords/arithmetic-progressions","display_name":"arithmetic progressions","score":0.8199999928474426},{"id":"https://openalex.org/keywords/additive-combinatorics","display_name":"additive combinatorics","score":0.7549999952316284},{"id":"https://openalex.org/keywords/finite-fields","display_name":"finite fields","score":0.39500001072883606}],"concepts":[{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.9610000252723694},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8379999995231628},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.8102999925613403},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.6723999977111816},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5802000164985657},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5662000179290771},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5024999976158142},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4401000142097473},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3750999867916107},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.2973000109195709},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.2515000104904175}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.10148","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10148","pdf_url":"https://arxiv.org/pdf/2609.10148","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.2609.10148","doi":"https://doi.org/10.48550/arxiv.2609.10148","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.10148","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:2609.10148","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10148","pdf_url":"https://arxiv.org/pdf/2609.10148","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/W7212123187.pdf","grobid_xml":"https://content.openalex.org/works/W7212123187.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7164397712","https://openalex.org/W7213298421","https://openalex.org/W7214045224","https://openalex.org/W4377371690","https://openalex.org/W2086777570","https://openalex.org/W4404314396","https://openalex.org/W4388907444","https://openalex.org/W2918152548","https://openalex.org/W2618878438","https://openalex.org/W3101373129"],"abstract_inverted_index":{"Let":[0],"$p$":[1],"be":[2,8],"a":[3,88,154],"prime":[4,130],"and":[5,57,61,66,86,125,152],"let":[6],"$A,B$":[7],"nonempty":[9],"subsets":[10],"of":[11,44,79,97,118],"the":[12,23,38,69,73,80,91,108,137,144,159,166],"cyclic":[13],"group":[14],"$\\mathbb{Z}_p$":[15],"with":[16,123],"$|A|\\neq":[17],"|B|$.":[18],"The":[19,41,94],"Alon--Nathanson--Ruzsa":[20],"theorem":[21,157],"gives":[22],"lower":[24],"bound":[25],"$|A\\rplus":[26,126],"B|\\ge":[27],"\\min\\{p,\\,|A|+|B|-2\\},$":[28],"where":[29],"$A\\rplus":[30],"B=\\{a+b:a\\in":[31],"A,\\":[32],"b\\in":[33],"B,\\":[34],"a\\neq":[35],"b\\}$":[36],"is":[37,82,100,140],"restricted":[39],"sumset.":[40],"inverse":[42,70,156],"problem":[43,71],"characterizing":[45],"all":[46],"critical":[47,120],"pairs":[48,121],"$(A,B)$":[49,122],"attaining":[50],"equality":[51],"was":[52],"posed":[53],"by":[54,148],"Alon,":[55],"Nathanson,":[56],"Ruzsa":[58],"in":[59,107],"1996":[60],"remains":[62],"open.":[63],"Recently,":[64],"Liu":[65],"Qian":[67],"solved":[68],"under":[72,158],"assumption":[74],"that":[75,103,136],"at":[76],"least":[77],"one":[78],"sets":[81],"an":[83],"arithmetic":[84],"progression,":[85],"proposed":[87],"conjecture":[89,105],"for":[90,128],"general":[92],"case.":[93,146],"main":[95],"purpose":[96],"this":[98,104],"paper":[99],"to":[101],"show":[102,135],"fails":[106],"boundary":[109,138],"case":[110,139],"$|A|+|B|=p$.":[111],"More":[112],"precisely,":[113],"we":[114,150],"construct":[115],"infinite":[116],"families":[117,134],"non-arithmetic":[119],"$|A|+|B|=p$":[124],"B|=p-2$":[127],"every":[129],"$p\\ge":[131],"11$.":[132],"These":[133],"fundamentally":[141],"different":[142],"from":[143],"non-boundary":[145],"Motivated":[147],"this,":[149],"formulate":[151],"prove":[153],"refined":[155],"natural":[160],"hypothesis":[161],"$|A|+|B|\\le":[162],"p-1$,":[163],"which":[164],"excludes":[165],"boundary.":[167]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-11T00:00:00"}