{"id":"https://openalex.org/W7214891758","doi":"https://doi.org/10.48550/arxiv.2609.38408","title":"Large gaps between Romanoff numbers","display_name":"Large gaps between Romanoff numbers","publication_year":2026,"publication_date":"2026-09-29","ids":{"openalex":"https://openalex.org/W7214891758","doi":"https://doi.org/10.48550/arxiv.2609.38408"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.38408","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.38408","pdf_url":"https://arxiv.org/pdf/2609.38408","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"},"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/2609.38408","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5011190707","display_name":"Artyom Olegovich Radomskii","orcid":"https://orcid.org/0000-0002-2675-2134","observed_orcids":["https://orcid.org/0000-0002-2675-2134"]},"institutions":[{"id":"https://openalex.org/I118501908","display_name":"National Research University Higher School of Economics","ror":"https://ror.org/055f7t516","country_code":"RU","type":"education","lineage":["https://openalex.org/I118501908"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"Radomskii, Artyom","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5011190707"],"corresponding_institution_ids":["https://openalex.org/I118501908"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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/T11166","display_name":"Analytic Number Theory Research","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":1.0,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":9.999999747378752e-05,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/divisor-sums","display_name":"divisor sums","score":0.6050000190734863},{"id":"https://openalex.org/keywords/sieve-methods","display_name":"sieve methods","score":0.5860000252723694}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8349999785423279},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.6607999801635742},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.6230999827384949},{"id":"https://openalex.org/C143732363","wikidata":"https://www.wikidata.org/wiki/Q485520","display_name":"Goldbach's conjecture","level":3,"score":0.6223999857902527},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5900999903678894},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.527999997138977},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.4586000144481659},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.45489999651908875},{"id":"https://openalex.org/C174072685","wikidata":"https://www.wikidata.org/wiki/Q1667469","display_name":"Prime power","level":3,"score":0.41830000281333923},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.41780000925064087},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.41760000586509705},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.41339999437332153},{"id":"https://openalex.org/C125257309","wikidata":"https://www.wikidata.org/wiki/Q131752","display_name":"Greatest common divisor","level":2,"score":0.4009999930858612},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.3808000087738037},{"id":"https://openalex.org/C203492994","wikidata":"https://www.wikidata.org/wiki/Q909669","display_name":"Divisor (algebraic geometry)","level":2,"score":0.36059999465942383},{"id":"https://openalex.org/C113429393","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime number","level":2,"score":0.35359999537467957},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.3483000099658966},{"id":"https://openalex.org/C203432157","wikidata":"https://www.wikidata.org/wiki/Q2498974","display_name":"Radical of an integer","level":4,"score":0.3345000147819519},{"id":"https://openalex.org/C164680029","wikidata":"https://www.wikidata.org/wiki/Q1055293","display_name":"Multiple","level":2,"score":0.33079999685287476},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.3296999931335449},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3280999958515167},{"id":"https://openalex.org/C93446932","wikidata":"https://www.wikidata.org/wiki/Q1972803","display_name":"Perfect power","level":2,"score":0.3249000012874603},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.32350000739097595},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.27399998903274536},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.26660001277923584},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.2646999955177307},{"id":"https://openalex.org/C163540240","wikidata":"https://www.wikidata.org/wiki/Q936614","display_name":"Almost prime","level":4,"score":0.25380000472068787}],"mesh":[],"locations_count":4,"locations":[{"id":"pmh:oai:arXiv.org:2609.38408","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.38408","pdf_url":"https://arxiv.org/pdf/2609.38408","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":"doi:10.48550/arxiv.2609.38408","doi":"https://doi.org/10.48550/arxiv.2609.38408","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.38408","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"},{"id":"doi:10.5281/zenodo.23064010","doi":"https://doi.org/10.5281/zenodo.23064010","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23064010","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.23064011","doi":"https://doi.org/10.5281/zenodo.23064011","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23064011","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":"pmh:oai:arXiv.org:2609.38408","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.38408","pdf_url":"https://arxiv.org/pdf/2609.38408","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":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320324261","display_name":"National Research University Higher School of Economics","ror":"https://ror.org/055f7t516"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214891758.pdf","grobid_xml":"https://content.openalex.org/works/W7214891758.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4416100789"],"related_works":["https://openalex.org/W4405994216","https://openalex.org/W1990484074","https://openalex.org/W4414682546","https://openalex.org/W7164525732","https://openalex.org/W2617906522","https://openalex.org/W4226302209","https://openalex.org/W2320148585","https://openalex.org/W2010462498","https://openalex.org/W7221541500","https://openalex.org/W2793709545"],"abstract_inverted_index":{"Questions":[0],"concerning":[1],"representations":[2],"of":[3,8,14,32,39,70,88,104,117,122,132],"integers":[4,40],"as":[5],"the":[6,19,37,50,75,80,89,99],"sum":[7],"a":[9,12,43,112],"prime":[10,94],"and":[11,23,27,56,95,106,155],"power":[13],"two":[15],"go":[16],"back":[17],"to":[18,28,73,115,147,158],"correspondence":[20],"between":[21],"Euler":[22],"Goldbach":[24],"in":[25,83],"1752":[26],"de":[29],"Polignac's":[30],"work":[31],"1849.":[33],"Romanoff":[34,64],"proved":[35],"that":[36],"set":[38],"admitting":[41],"such":[42],"representation":[44],"has":[45],"positive":[46],"lower":[47],"density.":[48],"Following":[49],"complementary":[51],"direction":[52],"studied":[53],"by":[54],"Kalmynin":[55,105],"Konyagin,":[57],"we":[58],"consider":[59],"long":[60],"intervals":[61],"containing":[62,85],"no":[63,86],"numbers.":[65],"We":[66],"use":[67],"truncated":[68],"products":[69],"divisor":[71],"sums":[72],"derive":[74],"bound":[76,100],"$G_{\\mathcal{R}}(X)\\gg\\log\\log":[77,101],"X$":[78,103],"for":[79,160],"longest":[81],"block":[82],"$[1,X]$":[84],"integer":[87,120,163],"form":[90],"$p+2^n$,":[91],"with":[92],"$p$":[93],"$n\\ge1$.":[96],"This":[97],"improves":[98],"X/\\log\\log\\log":[102],"Konyagin.":[107],"The":[108,151],"key":[109],"step":[110],"extends":[111],"translate":[113],"lemma":[114],"sets":[116],"$O(\\log":[118],"X)$":[119],"shifts":[121,133,146],"absolute":[123],"value":[124],"at":[125,136],"most":[126],"$X$:":[127],"auxiliary":[128],"primes":[129],"dividing":[130],"differences":[131],"are":[134],"removed":[135],"negligible":[137],"cost.":[138],"After":[139],"preliminary":[140],"sieving,":[141],"this":[142],"allows":[143],"all":[144],"remaining":[145],"be":[148],"covered":[149],"simultaneously.":[150],"argument":[152],"is":[153],"unconditional":[154],"also":[156],"applies":[157],"$p+a^n$":[159],"every":[161],"fixed":[162],"$a\\ge2$.":[164]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-01T00:00:00"}