{"id":"https://openalex.org/W2951414691","doi":"https://doi.org/10.48550/arxiv.1402.5775","title":"New sum-product estimates for real and complex numbers","display_name":"New sum-product estimates for real and complex numbers","publication_year":2014,"publication_date":"2014-02-24","ids":{"openalex":"https://openalex.org/W2951414691","doi":"https://doi.org/10.48550/arxiv.1402.5775","mag":"2951414691"},"language":"en","primary_location":{"id":"pmh:oai:real.mtak.hu:33579","doi":null,"is_oa":false,"landing_page_url":"http://real.mtak.hu/33579/1/1402.5775v1.pdf","pdf_url":"http://real.mtak.hu/33579/1/1402.5775v1.pdf","source":{"id":"https://openalex.org/S4306400081","display_name":"Repository of the Academy's Library (Library of the Hungarian Academy of Sciences)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4210140733","host_organization_name":"Library and Information Centre of the Hungarian Academy of Sciences","host_organization_lineage":["https://openalex.org/I4210140733"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Article"},"volume":null,"type":"article","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1402.5775","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051928221","display_name":"Antal Balog","orcid":"https://orcid.org/0000-0002-5202-2821","observed_orcids":["https://orcid.org/0000-0002-5202-2821"]},"institutions":[{"id":"https://openalex.org/I127817813","display_name":"HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics","ror":"https://ror.org/03vw74f64","country_code":"HU","type":"facility","lineage":["https://openalex.org/I127817813","https://openalex.org/I7597260"]},{"id":"https://openalex.org/I7597260","display_name":"Hungarian Academy of Sciences","ror":"https://ror.org/02ks8qq67","country_code":"HU","type":"government","lineage":["https://openalex.org/I7597260"]}],"countries":["HU"],"is_corresponding":false,"raw_author_name":"Balog, Antal","raw_affiliation_strings":["Alfr\u00e9d R\u00e9nyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary H-1053#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Alfr\u00e9d R\u00e9nyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary H-1053#TAB#","institution_ids":["https://openalex.org/I127817813","https://openalex.org/I7597260"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5091297941","display_name":"Oliver Roche\u2010Newton","orcid":"https://orcid.org/0000-0002-1640-3707","observed_orcids":["https://orcid.org/0000-0002-1640-3707"]},"institutions":[{"id":"https://openalex.org/I138211613","display_name":"Austrian Academy of Sciences","ror":"https://ror.org/03anc3s24","country_code":"AT","type":"government","lineage":["https://openalex.org/I138211613"]},{"id":"https://openalex.org/I4210158379","display_name":"Johann Radon Institute for Computational and Applied Mathematics","ror":"https://ror.org/05a94k872","country_code":"AT","type":"facility","lineage":["https://openalex.org/I138211613","https://openalex.org/I4210158379"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Roche-Newton, Oliver","raw_affiliation_strings":["Johann Radon Institute of Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria 4040#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Johann Radon Institute of Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria 4040#TAB#","institution_ids":["https://openalex.org/I138211613","https://openalex.org/I4210158379"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.5778,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.70675903,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"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":0.9092000126838684,"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":0.9092000126838684,"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":0.0568000003695488,"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":0.013299999758601189,"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/sum-product-problem","display_name":"sum-product problem","score":0.8299999833106995},{"id":"https://openalex.org/keywords/additive-combinatorics","display_name":"additive combinatorics","score":0.7950000166893005},{"id":"https://openalex.org/keywords/additive-energy","display_name":"additive energy","score":0.4309999942779541},{"id":"https://openalex.org/keywords/sumset-size","display_name":"sumset size","score":0.3959999978542328},{"id":"https://openalex.org/keywords/multiplicative-constants","display_name":"multiplicative constants","score":0.6029999852180481}],"concepts":[{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.8868510723114014},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.7152979373931885},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.6787766218185425},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6262896060943604},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6101690530776978},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.6050356030464172},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5979050397872925},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.44550788402557373},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3640793263912201},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10431396961212158},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.07669791579246521},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.04941016435623169},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"pmh:oai:real.mtak.hu:33579","doi":null,"is_oa":false,"landing_page_url":"http://real.mtak.hu/33579/1/1402.5775v1.pdf","pdf_url":"http://real.mtak.hu/33579/1/1402.5775v1.pdf","source":{"id":"https://openalex.org/S4306400081","display_name":"Repository of the Academy's Library (Library of the Hungarian Academy of Sciences)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4210140733","host_organization_name":"Library and Information Centre of the Hungarian Academy of Sciences","host_organization_lineage":["https://openalex.org/I4210140733"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:arXiv.org:1402.5775","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1402.5775","pdf_url":"https://arxiv.org/pdf/1402.5775","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:2951414691","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/1402.5775","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.760.5969","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.760.5969","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/1402.5775.pdf","raw_type":"text"},{"id":"doi:10.48550/arxiv.1402.5775","doi":"https://doi.org/10.48550/arxiv.1402.5775","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1402.5775","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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1402.5775","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1402.5775","pdf_url":"https://arxiv.org/pdf/1402.5775","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W1932410633","https://openalex.org/W1994132686","https://openalex.org/W2021383395","https://openalex.org/W2029408549","https://openalex.org/W2159774990","https://openalex.org/W2949267210","https://openalex.org/W2963755674"],"related_works":["https://openalex.org/W2963592743","https://openalex.org/W2146940934","https://openalex.org/W3042228208","https://openalex.org/W3136910307","https://openalex.org/W2963236425","https://openalex.org/W3140988650","https://openalex.org/W2953371808","https://openalex.org/W3142335819","https://openalex.org/W2617014456","https://openalex.org/W3209059903","https://openalex.org/W3119929907","https://openalex.org/W3166470924","https://openalex.org/W1992465788","https://openalex.org/W2962957985","https://openalex.org/W2963791499","https://openalex.org/W2509752846","https://openalex.org/W2963713484","https://openalex.org/W1990667482","https://openalex.org/W3099046895","https://openalex.org/W3048484789"],"abstract_inverted_index":{"A":[0],"variation":[1],"on":[2],"the":[3,79],"sum-product":[4],"problem":[5],"seeks":[6],"to":[7],"show":[8,37],"that":[9,38,51,63,72],"a":[10,54,84],"set":[11,42,86],"which":[12],"is":[13,49,60,70,83],"defined":[14],"by":[15],"additive":[16],"and":[17],"multiplicative":[18,93],"operations":[19],"will":[20],"always":[21],"be":[22],"large.":[23],"In":[24,34],"this":[25,32,57],"paper,":[26],"we":[27,36],"prove":[28],"new":[29],"results":[30],"of":[31,44,56,74,87],"type.":[33],"particular,":[35],"for":[39],"any":[40],"finite":[41,85],"$A$":[43,82],"positive":[45],"real":[46],"numbers,":[47,89],"it":[48,59,69],"true":[50],"$$\\left|\\left\\{\\frac{a+b}{c+d}:a,b,c,d\\in{A}\\right\\}\\right|\\geq{2|A|^2-1}.$$":[52],"As":[53],"consequence":[55],"result,":[58],"also":[61],"established":[62],"$$|4^{k-1}A^{(k)}|:=|\\underbrace{\\underbrace{A\\cdots{A}}_\\textrm{k":[64],"times}+\\cdots{+A\\cdots{A}}}_\\textrm{$4^{k-1}$":[65],"times}|\\geq{|A|^k}.$$":[66],"Later":[67],"on,":[68],"shown":[71],"both":[73],"these":[75],"bounds":[76],"hold":[77],"in":[78],"case":[80],"when":[81],"complex":[88],"although":[90],"with":[91],"smaller":[92],"constants.":[94]},"counts_by_year":[{"year":2014,"cited_by_count":1}],"updated_date":"2026-08-27T14:10:00.468798","created_date":"2025-10-10T00:00:00"}