{"id":"https://openalex.org/W3009417171","doi":"https://doi.org/10.1137/20m1325502","title":"Bounds of Trilinear and Trinomial Exponential Sums","display_name":"Bounds of Trilinear and Trinomial Exponential Sums","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3009417171","doi":"https://doi.org/10.1137/20m1325502","mag":"3009417171"},"language":"en","primary_location":{"id":"doi:10.1137/20m1325502","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1325502","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2003.03493","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Simon Macourt","orcid":"https://orcid.org/0000-0002-5136-0538"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Simon Macourt","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0002-5136-0538","affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Giorgis Petridis","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Giorgis Petridis","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Ilya D. Shkredov","orcid":"https://orcid.org/0000-0002-6445-8390"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ilya D. Shkredov","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0002-6445-8390","affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Igor E. Shparlinski","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Igor E. Shparlinski","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.0736,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.70700032,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"34","issue":"4","first_page":"2124","last_page":"2136"},"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":0.8296999931335449,"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":0.8296999931335449,"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":0.08179999887943268,"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/T12303","display_name":"Tensor decomposition and applications","score":0.010599999688565731,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational 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/multiplicative-function","display_name":"Multiplicative function","score":0.8263000249862671},{"id":"https://openalex.org/keywords/cartesian-product","display_name":"Cartesian product","score":0.7854999899864197},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.6430000066757202},{"id":"https://openalex.org/keywords/trinomial","display_name":"Trinomial","score":0.6150000095367432},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5097000002861023},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.46059998869895935}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9368000030517578},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.8263000249862671},{"id":"https://openalex.org/C65236422","wikidata":"https://www.wikidata.org/wiki/Q173740","display_name":"Cartesian product","level":2,"score":0.7854999899864197},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.6430000066757202},{"id":"https://openalex.org/C6791720","wikidata":"https://www.wikidata.org/wiki/Q1670509","display_name":"Trinomial","level":2,"score":0.6150000095367432},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5113999843597412},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5097000002861023},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.46059998869895935},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.43880000710487366},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43389999866485596},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3517000079154968},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.3296000063419342},{"id":"https://openalex.org/C112343008","wikidata":"https://www.wikidata.org/wiki/Q140744","display_name":"Arithmetic function","level":2,"score":0.31470000743865967},{"id":"https://openalex.org/C53638454","wikidata":"https://www.wikidata.org/wiki/Q4446020","display_name":"Exponential sum","level":3,"score":0.31369999051094055},{"id":"https://openalex.org/C157351203","wikidata":"https://www.wikidata.org/wiki/Q216906","display_name":"Multiplicative inverse","level":3,"score":0.2745000123977661},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.2721000015735626}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/20m1325502","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1325502","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2003.03493","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2003.03493","pdf_url":"https://arxiv.org/pdf/2003.03493","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2003.03493","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2003.03493","pdf_url":"https://arxiv.org/pdf/2003.03493","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":[],"awards":[{"id":"https://openalex.org/G3953265827","display_name":"Additive combinatorics, arithmetic algebraic geometry and finite fields","funder_award_id":"DP170100786","funder_id":"https://openalex.org/F4320334704","funder_display_name":"Australian Research Council"},{"id":"https://openalex.org/G4842867757","display_name":"Direct and Inverse Problems for Cardinality Questions in Additive Combinatorics","funder_award_id":"1723016","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5521849992","display_name":null,"funder_award_id":"075-15-2019-1614","funder_id":"https://openalex.org/F4320321912","funder_display_name":"Ministry of Education and Science of the Russian Federation"},{"id":"https://openalex.org/G6145688524","display_name":"\u6307\u6570\u548c\u3001\u7279\u5f81\u548c\u7684\u76f8\u6d88\u6027\u53ca\u76f8\u5173\u95ee\u9898\u7814\u7a76","funder_award_id":"11871317","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320310010","display_name":"University of Georgia","ror":"https://ror.org/00te3t702"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321912","display_name":"Ministry of Education and Science of the Russian Federation","ror":"https://ror.org/00ghqgy32"},{"id":"https://openalex.org/F4320334704","display_name":"Australian Research Council","ror":"https://ror.org/05mmh0f86"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":15,"referenced_works":["https://openalex.org/W138605413","https://openalex.org/W1738467033","https://openalex.org/W1965702545","https://openalex.org/W2002415417","https://openalex.org/W2026117048","https://openalex.org/W2031022397","https://openalex.org/W2586639504","https://openalex.org/W2750974893","https://openalex.org/W2771807507","https://openalex.org/W2963096283","https://openalex.org/W2963204401","https://openalex.org/W2963566185","https://openalex.org/W2963747036","https://openalex.org/W2964091994","https://openalex.org/W2964111809"],"related_works":[],"abstract_inverted_index":{"We":[0,31,57],"prove,":[1],"for":[2,80],"a":[3,9,60,70],"sufficiently":[4],"small":[5],"subset":[6],"$\\mathcal{A}$":[7],"of":[8,18,45,50,53,55,66,73],"prime":[10],"residue":[11],"field,":[12],"an":[13],"estimate":[14,62],"on":[15,36,41,63],"the":[16,21,42,48,64],"number":[17,44,65],"solutions":[19],"to":[20],"equation":[22],"$(a_1-a_2)(a_3-a_4)":[23],"=":[24],"(a_5-a_6)(a_7-a_8)$":[25],"with":[26,83],"all":[27,84],"variables":[28,85],"in":[29,69,86],"$\\mathcal{A}$.":[30,56],"then":[32],"derive":[33,77],"new":[34],"bounds":[35,79],"trilinear":[37,81],"exponential":[38],"sums":[39,82],"and":[40,76],"total":[43],"residues":[46],"equaling":[47],"product":[49,72],"two":[51],"differences":[52],"elements":[54],"also":[58],"prove":[59],"refined":[61],"collinear":[67],"triples":[68],"Cartesian":[71],"multiplicative":[74,87],"subgroups":[75],"stronger":[78],"subgroups.":[88]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2020-03-13T00:00:00"}