{"id":"https://openalex.org/W2949465255","doi":"https://doi.org/10.1137/140952004","title":"Variations on the Sum-Product Problem","display_name":"Variations on the Sum-Product Problem","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2949465255","doi":"https://doi.org/10.1137/140952004","mag":"2949465255"},"language":"en","primary_location":{"id":"doi:10.1137/140952004","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140952004","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":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Brendan Murphy","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Brendan Murphy","raw_affiliation_strings":["Pure Mathematics"],"affiliations":[{"raw_affiliation_string":"Pure Mathematics","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5091297941","display_name":"Oliver Roche\u2010Newton","orcid":"https://orcid.org/0000-0002-1640-3707"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Oliver Roche-Newton","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5071151408","display_name":"Ilya D. Shkredov","orcid":"https://orcid.org/0000-0002-6445-8390"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ilya Shkredov","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":8.8065,"has_fulltext":false,"cited_by_count":28,"citation_normalized_percentile":{"value":0.98663641,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"29","issue":"1","first_page":"514","last_page":"540"},"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":0.9997000098228455,"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":0.9997000098228455,"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":0.987500011920929,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9861000180244446,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/mathematics","display_name":"Mathematics","score":0.8417201042175293},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.7983383536338806},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7545747756958008},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.7256879210472107},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.720600426197052},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.5812171697616577},{"id":"https://openalex.org/keywords/element","display_name":"Element (criminal law)","score":0.4960809648036957},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.47803840041160583},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.43786996603012085},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4180615246295929},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09984943270683289},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.053679466247558594},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.04998844861984253}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8417201042175293},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7983383536338806},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7545747756958008},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.7256879210472107},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.720600426197052},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5812171697616577},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.4960809648036957},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.47803840041160583},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43786996603012085},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4180615246295929},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09984943270683289},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.053679466247558594},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.04998844861984253},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/140952004","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140952004","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:research-information.bris.ac.uk:openaire_cris_publications/5a89ec3e-7bf5-4969-9283-e0052ef5dd3f","is_oa":false,"landing_page_url":"https://research-information.bris.ac.uk/en/publications/5a89ec3e-7bf5-4969-9283-e0052ef5dd3f","pdf_url":null,"source":{"id":"https://openalex.org/S4306400895","display_name":"Bristol Research (University of Bristol)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I36234482","host_organization_name":"University of Bristol","host_organization_lineage":["https://openalex.org/I36234482"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Murphy, B, Roche-Newton, O & Shkredov, I D 2013, 'Variations on the Sum-Product Problem', SIAM Journal on Discrete Mathematics.","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.747.8259","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.747.8259","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1312.6438.pdf","raw_type":"text"},{"id":"pmh:oai:research-information.bris.ac.uk:publications/5a89ec3e-7bf5-4969-9283-e0052ef5dd3f","is_oa":false,"landing_page_url":"https://hdl.handle.net/1983/5a89ec3e-7bf5-4969-9283-e0052ef5dd3f","pdf_url":null,"source":{"id":"https://openalex.org/S7407055359","display_name":"Explore Bristol Research","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Murphy, B, Roche-Newton, O & Shkredov, I D 2013, 'Variations on the Sum-Product Problem', SIAM Journal on Discrete Mathematics.","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7418712141","display_name":"DTA - University of Reading","funder_award_id":"EP/K503125/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320320360","display_name":"University of Bristol","ror":"https://ror.org/0524sp257"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":36,"referenced_works":["https://openalex.org/W1585722514","https://openalex.org/W1655398400","https://openalex.org/W1836535392","https://openalex.org/W1965702545","https://openalex.org/W1976075351","https://openalex.org/W1985044931","https://openalex.org/W1992758844","https://openalex.org/W1994132686","https://openalex.org/W1998563847","https://openalex.org/W2021383395","https://openalex.org/W2026117048","https://openalex.org/W2029408549","https://openalex.org/W2045572795","https://openalex.org/W2063978664","https://openalex.org/W2073745474","https://openalex.org/W2086861401","https://openalex.org/W2101788557","https://openalex.org/W2103381743","https://openalex.org/W2108616329","https://openalex.org/W2141292457","https://openalex.org/W2144337080","https://openalex.org/W2159774990","https://openalex.org/W2166961358","https://openalex.org/W2167396159","https://openalex.org/W2167472711","https://openalex.org/W2289409611","https://openalex.org/W2949267210","https://openalex.org/W2950601047","https://openalex.org/W2963537911","https://openalex.org/W2963755674","https://openalex.org/W2963769834","https://openalex.org/W2964147088","https://openalex.org/W3102580705","https://openalex.org/W3103127587","https://openalex.org/W3123129672","https://openalex.org/W4205208713"],"related_works":["https://openalex.org/W3194471551","https://openalex.org/W4375956809","https://openalex.org/W1581373162","https://openalex.org/W2784911598","https://openalex.org/W4297950433","https://openalex.org/W4287023968","https://openalex.org/W4288279572","https://openalex.org/W1973978937","https://openalex.org/W4286909116","https://openalex.org/W4319874546"],"abstract_inverted_index":{"This":[0],"paper":[1],"considers":[2],"various":[3],"formulations":[4],"of":[5,26,69],"the":[6],"sum-product":[7,55],"problem.":[8],"It":[9],"is":[10,33,40],"shown":[11],"that,":[12],"for":[13,59,65],"a":[14,20,24,29,37,66],"finite":[15],"set":[16],"$A\\subset{\\mathbb{R}}$,":[17],"$|A(A+A)|\\gg{|A|^{\\frac{3}{2}+\\frac{1}{178}}},$":[18],"giving":[19],"partial":[21],"answer":[22],"to":[23,43],"conjecture":[25],"Balog.":[27],"In":[28],"similar":[30],"spirit,":[31],"it":[32],"established":[34],"that":[35,62],"$|A(A+A+A+A)|\\gg{\\frac{|A|^2}{\\log{|A|}}},$":[36],"bound":[38],"which":[39],"optimal":[41],"up":[42],"constant":[44],"and":[45,57],"logarithmic":[46],"factors.":[47],"We":[48],"also":[49],"prove":[50],"several":[51],"new":[52],"results":[53],"concerning":[54],"estimates":[56],"expanders,":[58],"example,":[60],"showing":[61],"$|A(A+a)|\\gg{|A|^{3/2}}$":[63],"holds":[64],"typical":[67],"element":[68],"$A$.":[70]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":4}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}