{"id":"https://openalex.org/W7204909495","doi":"https://doi.org/10.48550/arxiv.2608.30831","title":"Sharp discrete Hardy constants in dimensions three and four and strict upper bounds from dimension nine","display_name":"Sharp discrete Hardy constants in dimensions three and four and strict upper bounds from dimension nine","publication_year":2026,"publication_date":"2026-08-31","ids":{"openalex":"https://openalex.org/W7204909495","doi":"https://doi.org/10.48550/arxiv.2608.30831"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.30831","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30831","pdf_url":"https://arxiv.org/pdf/2608.30831","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/2608.30831","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5040965273","display_name":"Carlos Lizama","orcid":"https://orcid.org/0000-0002-9807-1100","observed_orcids":["https://orcid.org/0000-0002-9807-1100"]},"institutions":[{"id":"https://openalex.org/I10457146","display_name":"Universidad de Santiago de Chile","ror":"https://ror.org/02ma57s91","country_code":"CL","type":"education","lineage":["https://openalex.org/I10457146"]}],"countries":["CL"],"is_corresponding":true,"raw_author_name":"Lizama, Carlos","raw_affiliation_strings":["Departamento de Matem\u00e1tica y Ciencia de la Computaci\u00f3n , Facultad de Ciencia Universidad de Santiago de Chile Las Sophoras 173 , Estaci\u00f3n Central , Santiago , Chile"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1tica y Ciencia de la Computaci\u00f3n , Facultad de Ciencia Universidad de Santiago de Chile Las Sophoras 173 , Estaci\u00f3n Central , Santiago , Chile","institution_ids":["https://openalex.org/I10457146"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5040965273"],"corresponding_institution_ids":["https://openalex.org/I10457146"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":4,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9585000276565552,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9585000276565552,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.016499999910593033,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11564","display_name":"Mathematical Inequalities and Applications","score":0.005200000014156103,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/sharp-constants","display_name":"sharp constants","score":0.6759999990463257},{"id":"https://openalex.org/keywords/upper-bounds","display_name":"upper bounds","score":0.4620000123977661},{"id":"https://openalex.org/keywords/dimension-dependence","display_name":"dimension dependence","score":0.47600001096725464},{"id":"https://openalex.org/keywords/discrete-harmonic-functions","display_name":"discrete harmonic functions","score":0.35600000619888306}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8282999992370605},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6876999735832214},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.6140999794006348},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6057999730110168},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.47769999504089355},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4237000048160553},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4106000065803528},{"id":"https://openalex.org/C119322782","wikidata":"https://www.wikidata.org/wiki/Q2662236","display_name":"VC dimension","level":2,"score":0.3188000023365021},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.3021000027656555},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29899999499320984},{"id":"https://openalex.org/C46979015","wikidata":"https://www.wikidata.org/wiki/Q6666859","display_name":"Log sum inequality","level":3,"score":0.272599995136261},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.26989999413490295},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.2687999904155731}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.30831","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30831","pdf_url":"https://arxiv.org/pdf/2608.30831","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.2608.30831","doi":"https://doi.org/10.48550/arxiv.2608.30831","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.30831","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:2608.30831","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30831","pdf_url":"https://arxiv.org/pdf/2608.30831","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":[{"id":"https://openalex.org/F4320331146","display_name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","ror":"https://ror.org/05btm5233"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204909495.pdf","grobid_xml":"https://content.openalex.org/works/W7204909495.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W247697463","https://openalex.org/W2035925016","https://openalex.org/W2043985822","https://openalex.org/W2069528055","https://openalex.org/W2560406243","https://openalex.org/W2564522201","https://openalex.org/W2565466329","https://openalex.org/W2751955921","https://openalex.org/W2802008906","https://openalex.org/W2962857095","https://openalex.org/W2963111754","https://openalex.org/W2971788505","https://openalex.org/W3014169485","https://openalex.org/W3139954217","https://openalex.org/W3191131957","https://openalex.org/W4229706427","https://openalex.org/W4315567937","https://openalex.org/W4366823519","https://openalex.org/W4388831659","https://openalex.org/W4392604545","https://openalex.org/W4400324777","https://openalex.org/W4406758564","https://openalex.org/W4411957555","https://openalex.org/W4413812809","https://openalex.org/W4416103210","https://openalex.org/W4417251175","https://openalex.org/W4417256335","https://openalex.org/W7118943058","https://openalex.org/W7169496690","https://openalex.org/W7171533864","https://openalex.org/W7204437108"],"related_works":["https://openalex.org/W7220661842","https://openalex.org/W7099210458","https://openalex.org/W3043154566","https://openalex.org/W2508476950","https://openalex.org/W7169496690","https://openalex.org/W7204437108","https://openalex.org/W2809756301","https://openalex.org/W7204868704","https://openalex.org/W7099110118","https://openalex.org/W7099123104"],"abstract_inverted_index":{"For":[0],"$N\\ge3$,":[1],"let":[2],"$C(N)$":[3],"be":[4],"the":[5,9,24,38],"optimal":[6],"constant":[7],"in":[8,33],"nearest-neighbour":[10],"Hardy":[11],"inequality":[12],"on":[13],"$\\mathbb":[14],"Z^N$":[15],"with":[16],"$u(0)=0$,":[17],"and":[18,36,42],"set":[19],"$A_N=(N-2)^2/4$.":[20],"We":[21],"prove":[22],"that":[23],"continuum":[25],"coefficient":[26],"remains":[27],"an":[28],"upper":[29],"bound,":[30],"$C(N)\\le":[31],"A_N$,":[32],"every":[34],"dimension,":[35],"determine":[37],"exact":[39],"values":[40],"$C(3)=A_3=1/4$":[41],"$C(4)=A_4=1$.":[43],"In":[44],"higher":[45],"dimensions":[46],"we":[47],"show":[48],"$C(N)":[49]},"counts_by_year":[{"year":2026,"cited_by_count":4}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-02T00:00:00"}