{"id":"https://openalex.org/W7163336205","doi":"https://doi.org/10.48550/arxiv.2606.03627","title":"On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications","display_name":"On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications","publication_year":2026,"publication_date":"2026-06-02","ids":{"openalex":"https://openalex.org/W7163336205","doi":"https://doi.org/10.48550/arxiv.2606.03627"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2606.03627","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.03627","pdf_url":"https://arxiv.org/pdf/2606.03627","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/2606.03627","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5001828232","display_name":"Thang Luong Pham","orcid":"https://orcid.org/0000-0001-7600-7450","observed_orcids":["https://orcid.org/0000-0001-7600-7450","https://orcid.org/0000-0002-8514-0808"]},"institutions":[{"id":"https://openalex.org/I149594827","display_name":"Xidian University","ror":"https://ror.org/05s92vm98","country_code":"CN","type":"education","lineage":["https://openalex.org/I149594827"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Pham, Thang","raw_affiliation_strings":["Institute of Mathematics and Interdisciplinary Sciences , Xidian University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics and Interdisciplinary Sciences , Xidian University","institution_ids":["https://openalex.org/I149594827"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5081410471","display_name":"Boqing Xue","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I30809798","display_name":"ShanghaiTech University","ror":"https://ror.org/030bhh786","country_code":"CN","type":"education","lineage":["https://openalex.org/I30809798"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xue, Boqing","raw_affiliation_strings":["Institute of Mathematical Sciences , ShanghaiTech University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematical Sciences , ShanghaiTech University","institution_ids":["https://openalex.org/I30809798"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":3,"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":0.6259999871253967,"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":0.6259999871253967,"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/T11049","display_name":"Advanced Harmonic Analysis Research","score":0.1559000015258789,"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/T11166","display_name":"Analytic Number Theory Research","score":0.11909999698400497,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/finite-fields","display_name":"finite fields","score":0.9440000057220459},{"id":"https://openalex.org/keywords/hecke-operators","display_name":"Hecke operators","score":0.7590000033378601},{"id":"https://openalex.org/keywords/orthogonal-decomposition","display_name":"orthogonal decomposition","score":0.8809999823570251},{"id":"https://openalex.org/keywords/distance-sets","display_name":"distance sets","score":0.5550000071525574},{"id":"https://openalex.org/keywords/four-dimensional-euclidean-space","display_name":"four-dimensional Euclidean space","score":0.5139999985694885}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7628999948501587},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.6114000082015991},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.6000999808311462},{"id":"https://openalex.org/C2776190703","wikidata":"https://www.wikidata.org/wiki/Q488148","display_name":"Slicing","level":2,"score":0.5763999819755554},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.550599992275238},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.5113999843597412},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4189999997615814},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.4171999990940094},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.40790000557899475},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4068000018596649},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3537999987602234},{"id":"https://openalex.org/C114410712","wikidata":"https://www.wikidata.org/wiki/Q973321","display_name":"Hypersurface","level":2,"score":0.3271999955177307},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.2962999939918518},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2962000072002411},{"id":"https://openalex.org/C81265862","wikidata":"https://www.wikidata.org/wiki/Q225527","display_name":"Ramification","level":2,"score":0.2946999967098236},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2802000045776367},{"id":"https://openalex.org/C138268822","wikidata":"https://www.wikidata.org/wiki/Q1051925","display_name":"Resolution (logic)","level":2,"score":0.2773999869823456},{"id":"https://openalex.org/C2778664614","wikidata":"https://www.wikidata.org/wiki/Q575597","display_name":"Spherical cap","level":2,"score":0.2718000113964081},{"id":"https://openalex.org/C2777229593","wikidata":"https://www.wikidata.org/wiki/Q1531713","display_name":"Global field","level":2,"score":0.25999999046325684},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.25220000743865967}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2606.03627","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.03627","pdf_url":"https://arxiv.org/pdf/2606.03627","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2606.03627","doi":"https://doi.org/10.48550/arxiv.2606.03627","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.03627","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:2606.03627","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.03627","pdf_url":"https://arxiv.org/pdf/2606.03627","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/W7163336205.pdf","grobid_xml":"https://content.openalex.org/works/W7163336205.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W2565402110","https://openalex.org/W4366400270","https://openalex.org/W4396913558","https://openalex.org/W4400611981","https://openalex.org/W7103142588","https://openalex.org/W7133316105","https://openalex.org/W7138893547"],"related_works":["https://openalex.org/W2811406629","https://openalex.org/W7223217034","https://openalex.org/W3167629974","https://openalex.org/W2898267824","https://openalex.org/W3013957954","https://openalex.org/W2891291067","https://openalex.org/W3008010078","https://openalex.org/W7215080244","https://openalex.org/W4399115867","https://openalex.org/W3181275989"],"abstract_inverted_index":{"Let":[0],"$p$":[1,27],"be":[2],"an":[3,43,88],"odd":[4],"prime.":[5],"We":[6,46],"prove":[7],"the":[8,56,60,63,80,94],"sharp":[9,57],"extension":[10],"estimate":[11],"$R_{S_j}^*(2\\to":[12],"r)\\lesssim_r":[13],"1$":[14],"for":[15,75,103],"every":[16,23,77,104,117],"sphere":[17],"of":[18,62,107],"nonzero":[19],"radius":[20],"$S_j\\subseteq\\mathbb{F}_p^4$":[21],"and":[22,28,42,70],"$r\\geq3$,":[24],"uniformly":[25],"in":[26,84,93],"$j$.":[29],"The":[30],"proof":[31],"combines":[32],"arithmetic":[33],"Hecke":[34],"operator":[35],"bounds":[36],"with":[37,87,120],"a":[38,49],"refined":[39],"fourth":[40],"moment":[41],"orthogonal":[44],"decomposition.":[45],"also":[47],"formulate":[48],"localized":[50,99],"spherical":[51],"restriction/extension":[52],"conjecture":[53,67],"that":[54],"predicts":[55],"dependence":[58],"on":[59],"size":[61,95,108],"physical":[64],"support.":[65],"This":[66],"remains":[68],"open":[69],"would":[71],"yield":[72],"distance":[73],"estimates":[74,100],"almost":[76,116],"pin":[78],"at":[79,109],"conjectured":[81],"Erd\u0151s--Falconer":[82],"exponent":[83],"four":[85],"dimensions,":[86],"arbitrarily":[89],"small":[90],"power":[91],"loss":[92],"hypothesis.":[96],"Unconditionally,":[97],"our":[98],"imply":[101],"that,":[102],"$\\varepsilon>0$,":[105],"sets":[106],"least":[110],"$p^{7/3+\\varepsilon}$":[111],"determine":[112],"$(1+o_\\varepsilon(1))p$":[113],"distances":[114],"from":[115],"pin,":[118],"together":[119],"asymmetric":[121],"two-set":[122],"versions.":[123]},"counts_by_year":[{"year":2026,"cited_by_count":3}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-06-04T00:00:00"}