{"id":"https://openalex.org/W7225323404","doi":"https://doi.org/10.5281/zenodo.23976156","title":"A conservation law for the first moment in Graham's rearrangement problem","display_name":"A conservation law for the first moment in Graham's rearrangement problem","publication_year":2026,"publication_date":"2026-11-05","ids":{"openalex":"https://openalex.org/W7225323404","doi":"https://doi.org/10.5281/zenodo.23976156"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23976156","doi":"https://doi.org/10.5281/zenodo.23976156","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23976156","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","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23976156","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5144200242","display_name":"Zeno Lee","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Lee, Zeno","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5144200242"],"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.6327999830245972,"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":0.6327999830245972,"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":0.20389999449253082,"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":0.13459999859333038,"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":0.8349999785423279},{"id":"https://openalex.org/keywords/partial-sums","display_name":"partial sums","score":0.6660000085830688},{"id":"https://openalex.org/keywords/conservation-laws","display_name":"conservation laws","score":0.6240000128746033},{"id":"https://openalex.org/keywords/first-moment","display_name":"first moment","score":0.7730000019073486},{"id":"https://openalex.org/keywords/anti-concentration","display_name":"anti-concentration","score":0.7319999933242798},{"id":"https://openalex.org/keywords/extremal-constants","display_name":"extremal constants","score":0.5260000228881836},{"id":"https://openalex.org/keywords/random-ordering","display_name":"random ordering","score":0.5080000162124634},{"id":"https://openalex.org/keywords/extremal-set-theory","display_name":"extremal set theory","score":0.4620000123977661},{"id":"https://openalex.org/keywords/combinatorial-number-theory","display_name":"combinatorial number theory","score":0.4860000014305115},{"id":"https://openalex.org/keywords/extremal-graph-theory","display_name":"extremal graph theory","score":0.3310000002384186},{"id":"https://openalex.org/keywords/extremal-hypergraph-theory","display_name":"extremal hypergraph theory","score":0.5099999904632568}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7979000210762024},{"id":"https://openalex.org/C50742451","wikidata":"https://www.wikidata.org/wiki/Q1530388","display_name":"Equidistributed sequence","level":2,"score":0.7781000137329102},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.7346000075340271},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5453000068664551},{"id":"https://openalex.org/C4285052","wikidata":"https://www.wikidata.org/wiki/Q3406268","display_name":"Moment problem","level":3,"score":0.5002999901771545},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.46059998869895935},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.4602999985218048},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.42890000343322754},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.3481000065803528},{"id":"https://openalex.org/C106195933","wikidata":"https://www.wikidata.org/wiki/Q7847935","display_name":"Truncation (statistics)","level":2,"score":0.3391000032424927},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3303000032901764},{"id":"https://openalex.org/C61062188","wikidata":"https://www.wikidata.org/wiki/Q835065","display_name":"Second moment of area","level":2,"score":0.3179999887943268},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.29760000109672546},{"id":"https://openalex.org/C35651441","wikidata":"https://www.wikidata.org/wiki/Q625303","display_name":"Independence (probability theory)","level":2,"score":0.2858999967575073},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.27649998664855957}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.5281/zenodo.23976156","doi":"https://doi.org/10.5281/zenodo.23976156","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23976156","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.23976157","doi":"https://doi.org/10.5281/zenodo.23976157","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23976157","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.24002893","doi":"https://doi.org/10.5281/zenodo.24002893","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24002893","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.24051508","doi":"https://doi.org/10.5281/zenodo.24051508","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24051508","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.23976156","doi":"https://doi.org/10.5281/zenodo.23976156","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23976156","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W7119728444","https://openalex.org/W7223820326","https://openalex.org/W7225117671"],"related_works":["https://openalex.org/W7172105904","https://openalex.org/W7225117671","https://openalex.org/W7202275859","https://openalex.org/W7077896492","https://openalex.org/W4416971525","https://openalex.org/W7223820326","https://openalex.org/W7221805267","https://openalex.org/W2032983827","https://openalex.org/W4414990888","https://openalex.org/W7127929208"],"abstract_inverted_index":{"Let":[0],"$G$":[1],"be":[2,26],"a":[3,36,60],"finite":[4],"abelian":[5,274],"group":[6],"of":[7,30,32,35,39,84,87,143,193],"order":[8,192],"$n$":[9,85],"and":[10,55,81,86,107,148,166,179,183,204,286],"$A":[11,258],"\\subseteq":[12],"G":[13],"\\setminus":[14,261,280],"\\{0\\}$":[15,281],"with":[16,212],"$|A|":[17],"=":[18,45,76,125,137,218,256,259,270,284],"t$.":[19],"For":[20],"$g":[21,44],"\\in":[22,73],"G$":[23],"let":[24],"$U_g(A)$":[25],"the":[27,48,65,98,113,116,140,172,188,191,201,224,234,265,300],"expected":[28],"number":[29],"pairs":[31],"partial":[33],"sums":[34],"random":[37],"ordering":[38],"$A$":[40,80],"differing":[41],"by":[42,276],"$-g$;":[43],"0$":[46],"gives":[47,214,282],"first":[49,67,117],"moment":[50,68,118],"for":[51,78],"Graham's":[52],"rearrangement":[53],"problem,":[54],"$U_0(A)":[56,215],"<":[57],"1$":[58],"certifies":[59],"valid":[61],"ordering.":[62],"We":[63],"prove":[64],"total":[66],"is":[69,119,245,251,292,302],"an":[70,103],"invariant:":[71],"$\\sum_{g":[72],"G}":[74],"U_g(A)":[75,124],"\\binom{t-1}{2}$":[77,93],"every":[79],"$G$,":[82],"independently":[83],"$A$.":[88],"At":[89],"least":[90],"$n":[91],"-":[92,229,231],"targets":[94],"are":[95,162],"certified,":[96],"recovering":[97],"observed":[99],"$\\sqrt{2p}$":[100],"threshold":[101],"as":[102],"identity.":[104],"Feeding":[105],"Pham":[106],"Sauermann's":[108],"fixed-size":[109,185],"anticoncentration":[110],"theorem":[111],"into":[112],"identity":[114],"shows":[115],"equidistributed":[120],"over":[121,187,272],"$\\mathbb{Z}_p$,":[122],"$\\max_g":[123],"\\frac{1}{p}\\binom{t-1}{2}":[126],"+":[127,221],"O(\\sqrt{t":[128],"\\log":[129,133,239],"t})$,":[130],"so":[131],"$O(\\sqrt{t":[132],"t})$":[134],"when":[135],"$t":[136,228,269],"o(\\sqrt":[138],"p)$;":[139],"Kneserized":[141],"estimate":[142,170,186],"Costa,":[144],"Della":[145],"Fiore,":[146],"Feng":[147],"Liu":[149],"extends":[150],"this":[151],"to":[152,157,174],"composite":[153],"cyclic":[154],"moduli,":[155],"subject":[156],"their":[158],"dichotomy.":[159],"Two":[160,263],"upgrades":[161],"incorporated:":[163],"Nguyen,":[164],"Tran":[165],"Vu's":[167,184],"announced":[168],"two-sided":[169],"improves":[171],"error":[173],"$O(\\sqrt":[175],"t)$":[176],"(proof":[177],"deferred),":[178],"Berger,":[180],"Berkowitz,":[181],"Devlin":[182],"reals":[189],"settles":[190],"magnitude":[194],"in":[195,237,295],"our":[196],"open":[197],"slice-anticoncentration":[198],"question,":[199],"leaving":[200],"sharp":[202],"constant":[203,236],"extremiser":[205],"open.":[206],"Combining":[207],"$z":[208,255],"N_z(A,g)":[209],"\\le":[210,216],"\\binom{t}{z-1}$":[211],"complementation":[213],"B(t)":[217],"(\\log":[219],"2":[220],"o(1))t$,":[222],"improving":[223],"recorded":[225],"worst":[226],"case":[227],"1":[230],"H_{t-1}$,":[232],"trapping":[233],"extremal":[235,293],"$[1/2,":[238],"2]$;":[240],"we":[241],"ask":[242],"whether":[243],"it":[244],"$1/2$.":[246],"Each":[247],"inequality":[248],"behind":[249],"$B(t)$":[250],"sharp,":[252],"attained":[253],"at":[254,268],"3$,":[257],"\\mathbb{F}_2^m":[260],"\\{0\\}$.":[262],"corrections:":[264],"method":[266],"fails":[267],"5$":[271],"general":[273],"groups":[275],"strict":[277],"excess":[278],"($\\mathbb{Z}_6":[279],"$U_0":[283],"6/5$),":[285],"$\\{\\pm":[287],"1,":[288],"\\dots,":[289],"\\pm":[290],"k\\}$":[291],"only":[294],"its":[296],"minimal":[297],"ambient":[298],"group:":[299],"obstruction":[301],"density,":[303],"not":[304],"symmetry.":[305]},"counts_by_year":[],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-11-06T00:00:00"}
