{"id":"https://openalex.org/W7223914927","doi":"https://doi.org/10.5281/zenodo.23680960","title":"Distinct distances in general position: an explicit improvement of the bound n/3","display_name":"Distinct distances in general position: an explicit improvement of the bound n/3","publication_year":2026,"publication_date":"2026-10-27","ids":{"openalex":"https://openalex.org/W7223914927","doi":"https://doi.org/10.5281/zenodo.23680960"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23680960","doi":"https://doi.org/10.5281/zenodo.23680960","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23680960","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23680960","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5132755455","display_name":"Adam McKenna","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"McKenna, Adam","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5132755455"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.3700999915599823,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.3700999915599823,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.29030001163482666,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.13179999589920044,"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/distinct-distances","display_name":"distinct distances","score":0.8379999995231628},{"id":"https://openalex.org/keywords/points-in-general-position","display_name":"points in general position","score":0.8029999732971191},{"id":"https://openalex.org/keywords/distance-distribution","display_name":"distance distribution","score":0.32499998807907104},{"id":"https://openalex.org/keywords/linear-inequalities","display_name":"linear inequalities","score":0.5809999704360962},{"id":"https://openalex.org/keywords/lean-4-formalization","display_name":"Lean 4 formalization","score":0.8450000286102295},{"id":"https://openalex.org/keywords/automated-theorem-proving","display_name":"automated theorem proving","score":0.5059999823570251}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7804999947547913},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5485000014305115},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.5317000150680542},{"id":"https://openalex.org/C96865113","wikidata":"https://www.wikidata.org/wiki/Q2946816","display_name":"Certificate","level":2,"score":0.49959999322891235},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.489300012588501},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48429998755455017},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.43309998512268066},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.3783999979496002},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.3718999922275543},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.3481000065803528},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.2948000133037567},{"id":"https://openalex.org/C103278499","wikidata":"https://www.wikidata.org/wiki/Q254465","display_name":"Similarity (geometry)","level":3,"score":0.28690001368522644},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.2840999960899353},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.2782000005245209},{"id":"https://openalex.org/C96146094","wikidata":"https://www.wikidata.org/wiki/Q609057","display_name":"Unification","level":2,"score":0.2750999927520752},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.2750999927520752},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.25999999046325684},{"id":"https://openalex.org/C189430467","wikidata":"https://www.wikidata.org/wiki/Q7293293","display_name":"Ranking (information retrieval)","level":2,"score":0.2581999897956848}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.5281/zenodo.23680960","doi":"https://doi.org/10.5281/zenodo.23680960","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23680960","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23680961","doi":"https://doi.org/10.5281/zenodo.23680961","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23680961","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23681146","doi":"https://doi.org/10.5281/zenodo.23681146","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23681146","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.23706854","doi":"https://doi.org/10.5281/zenodo.23706854","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23706854","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"doi:10.5281/zenodo.23680960","doi":"https://doi.org/10.5281/zenodo.23680960","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23680960","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3099831362","https://openalex.org/W7221611116","https://openalex.org/W4236443031","https://openalex.org/W7096342531","https://openalex.org/W2591626202","https://openalex.org/W7099836553","https://openalex.org/W7098496959","https://openalex.org/W2012422741","https://openalex.org/W2027914059","https://openalex.org/W7124273001"],"abstract_inverted_index":{"Let":[0],"P":[1,28,38],"be":[2],"a":[3,25,71,78],"set":[4,52,72],"of":[5,27,35,84,94,119,190,227],"n":[6,54],"points":[7,14,19,34,57],"in":[8,77,148],"the":[9,92,95,116,120,138,144,153,163,169,204,206,212,219,222,225,228,240,247,250,255,257,260,276,278,280,284,290],"plane":[10],"with":[11,53,73,102,143,151,181,221,246],"no":[12,17],"three":[13,32],"collinear":[15],"and":[16,100,162,173,185,200,208,215,217,243,249,259,267,283],"four":[18],"concyclic.":[20],"Every":[21],"circle":[22],"centred":[23],"at":[24,30,40],"point":[26],"holds":[29],"most":[31],"other":[33],"P,":[36],"so":[37],"determines":[39,58],"least":[41],"(n":[42],"\u2212":[43],"1)/3":[44],"distinct":[45,64],"distances.":[46,65],"We":[47],"prove":[48],"that":[49,70,91,115,252],"every":[50,125],"such":[51,114],"\u2265":[55],"10\u2074":[56],"more":[59],"than":[60],"(1/3":[61],"+":[62],"1/40)n":[63],"The":[66,140,175,230,264],"proof":[67],"first":[68],"shows":[69],"few":[74],"distances":[75,97],"lies":[76],"thin":[79],"annulus":[80],"around":[81],"its":[82,209],"centre":[83],"mass.":[85],"It":[86],"then":[87],"derives":[88],"linear":[89],"inequalities":[90,121],"distribution":[93],"pairwise":[96],"must":[98],"satisfy,":[99],"closes":[101],"an":[103,182],"exact":[104,131],"rational":[105,112],"Farkas":[106],"certificate":[107,155,207],"over":[108],"60":[109,154],"cases:":[110],"nonnegative":[111],"weights":[113],"weighted":[117],"sum":[118],"is":[122,146,177],"false":[123],"for":[124,179],"possible":[126],"distance":[127],"profile.":[128],"An":[129],"independent":[130],"verifier":[132],"(Python":[133],"standard":[134],"library":[135],"only)":[136],"checks":[137],"certificate.":[139],"complete":[141],"proof,":[142],"certificate,":[145,256,279],"formalized":[147],"Lean":[149,213,261,281],"4":[150,262],"Mathlib;":[152],"cases":[156],"are":[157,269,287],"checked":[158],"by":[159],"decide":[160],"+kernel,":[161],"final":[164],"theorem":[165],"depends":[166],"only":[167],"on":[168],"axioms":[170],"propext,":[171],"Classical.choice":[172],"Quot.sound.":[174],"paper":[176,241,265],"written":[178],"readers":[180],"undergraduate":[183],"background":[184],"has":[186],"11":[187],"figures.":[188],"Use":[189],"AI:":[191],"AI":[192],"agents,":[193],"among":[194],"them":[195],"ChatGPT":[196],"6.0":[197],"Astra":[198],"(OpenAI)":[199],"Claude":[201],"(Anthropic),":[202],"developed":[203],"arguments,":[205],"verifier,":[210,258,277],"wrote":[211,218],"definitions":[214],"proofs,":[216],"text":[220,266],"author,":[223],"under":[224,271,289],"direction":[226],"author.":[229],"results":[231],"have":[232],"not":[233],"been":[234],"peer":[235],"reviewed.":[236],"This":[237],"record":[238],"contains":[239],"(PDF":[242],"LaTeX":[244],"source,":[245],"figures":[248,268],"programs":[251,286],"draw":[253],"them),":[254],"formalization.":[263],"licensed":[270,288],"CC":[272],"BY":[273],"4.0":[274],"(paper/LICENSE);":[275],"sources":[282],"figure":[285],"Apache":[291],"License":[292],"2.0":[293],"(LICENSE).":[294]},"counts_by_year":[],"updated_date":"2026-10-29T09:56:32.521200","created_date":"2026-10-28T00:00:00"}
