{"id":"https://openalex.org/W4416256366","doi":"https://doi.org/10.48550/arxiv.2509.20181","title":"Signed null sequences and Hausdorff dimension","display_name":"Signed null sequences and Hausdorff dimension","publication_year":2025,"publication_date":"2025-09-24","ids":{"openalex":"https://openalex.org/W4416256366","doi":"https://doi.org/10.48550/arxiv.2509.20181"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2509.20181","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2509.20181","pdf_url":"https://arxiv.org/pdf/2509.20181","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"},"parent_works":[],"lineage":["https://openalex.org/W4416256366"],"child_works_count":0,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2509.20181","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038614472","display_name":"Rich\u00e1rd Balka","orcid":"https://orcid.org/0000-0002-5633-4806","observed_orcids":["https://orcid.org/0000-0002-5633-4806"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Balka, Rich\u00e1rd","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087874914","display_name":"Korn\u00e9lia H\u00e9ra","orcid":"https://orcid.org/0000-0002-4540-0054","observed_orcids":["https://orcid.org/0000-0002-4540-0054"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"H\u00e9ra, Korn\u00e9lia","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5065864090","display_name":"Gergely Kiss","orcid":"https://orcid.org/0000-0001-5517-5148","observed_orcids":["https://orcid.org/0000-0001-5517-5148"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kiss, Gergely","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":null,"has_fulltext":true,"cited_by_count":0,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.4814000129699707,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.4814000129699707,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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":0.19519999623298645,"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/T13234","display_name":"p-Adic Analysis and Mathematical Physics","score":0.10930000245571136,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/hausdorff-dimension","display_name":"Hausdorff dimension","score":0.8939999938011169},{"id":"https://openalex.org/keywords/convergence-of-series","display_name":"convergence of series","score":0.6140000224113464},{"id":"https://openalex.org/keywords/sign-sequences","display_name":"sign sequences","score":0.8349999785423279},{"id":"https://openalex.org/keywords/ultrametric-topology","display_name":"ultrametric topology","score":0.9520000219345093},{"id":"https://openalex.org/keywords/linear-independence","display_name":"linear independence","score":0.5580000281333923},{"id":"https://openalex.org/keywords/extra-dimensions","display_name":"extra dimensions","score":0.3529999852180481}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8644000291824341},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.7741000056266785},{"id":"https://openalex.org/C115311070","wikidata":"https://www.wikidata.org/wiki/Q5347255","display_name":"Effective dimension","level":3,"score":0.6011999845504761},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5673999786376953},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5580999851226807},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.5533999800682068},{"id":"https://openalex.org/C141898687","wikidata":"https://www.wikidata.org/wiki/Q1501997","display_name":"Hausdorff distance","level":2,"score":0.5529999732971191},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5130000114440918},{"id":"https://openalex.org/C203763787","wikidata":"https://www.wikidata.org/wiki/Q371029","display_name":"Null (SQL)","level":2,"score":0.49889999628067017},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.4812000095844269},{"id":"https://openalex.org/C73225184","wikidata":"https://www.wikidata.org/wiki/Q1591095","display_name":"Hausdorff measure","level":3,"score":0.4683000147342682},{"id":"https://openalex.org/C28235433","wikidata":"https://www.wikidata.org/wiki/Q5277259","display_name":"Dimension function","level":3,"score":0.4650999903678894},{"id":"https://openalex.org/C101597101","wikidata":"https://www.wikidata.org/wiki/Q2555655","display_name":"Minkowski\u2013Bouligand dimension","level":4,"score":0.4634999930858612},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44690001010894775},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.4399999976158142},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.42320001125335693},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.40290001034736633},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.39879998564720154},{"id":"https://openalex.org/C84496893","wikidata":"https://www.wikidata.org/wiki/Q1135782","display_name":"Urysohn and completely Hausdorff spaces","level":4,"score":0.39730000495910645},{"id":"https://openalex.org/C71287196","wikidata":"https://www.wikidata.org/wiki/Q847204","display_name":"Limit of a sequence","level":3,"score":0.3377000093460083},{"id":"https://openalex.org/C143924541","wikidata":"https://www.wikidata.org/wiki/Q258374","display_name":"Outer measure","level":5,"score":0.33390000462532043},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.31859999895095825},{"id":"https://openalex.org/C61067352","wikidata":"https://www.wikidata.org/wiki/Q1897429","display_name":"Ultrametric space","level":3,"score":0.2955999970436096},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.28999999165534973},{"id":"https://openalex.org/C2777478746","wikidata":"https://www.wikidata.org/wiki/Q5358910","display_name":"Elementary proof","level":2,"score":0.28459998965263367},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.2727999985218048},{"id":"https://openalex.org/C80661125","wikidata":"https://www.wikidata.org/wiki/Q5404299","display_name":"Packing dimension","level":5,"score":0.257999986410141},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.25369998812675476}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2509.20181","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2509.20181","pdf_url":"https://arxiv.org/pdf/2509.20181","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.2509.20181","doi":"https://doi.org/10.48550/arxiv.2509.20181","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2509.20181","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:2509.20181","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2509.20181","pdf_url":"https://arxiv.org/pdf/2509.20181","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":[{"id":"https://openalex.org/G1521016025","display_name":"EXC 2047: Hausdorff Center for Mathematics (HCM)","funder_award_id":"390685813","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G350592136","display_name":null,"funder_award_id":"146922","funder_id":"https://openalex.org/F4320336675","funder_display_name":"National Research, Development and Innovation Office"},{"id":"https://openalex.org/G4093683979","display_name":null,"funder_award_id":"SFB1060","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G5184470010","display_name":null,"funder_award_id":"146922","funder_id":"https://openalex.org/F4320326762","funder_display_name":"Nemzeti Kutat\u00e1si Fejleszt\u00e9si \u00e9s Innov\u00e1ci\u00f3s Hivatal"},{"id":"https://openalex.org/G8847963564","display_name":null,"funder_award_id":"GZ 2047/1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320322192","display_name":"Magyar Tudom\u00e1nyos Akad\u00e9mia","ror":"https://ror.org/02ks8qq67"},{"id":"https://openalex.org/F4320326762","display_name":"Nemzeti Kutat\u00e1si Fejleszt\u00e9si \u00e9s Innov\u00e1ci\u00f3s Hivatal","ror":null},{"id":"https://openalex.org/F4320336675","display_name":"National Research, Development and Innovation Office","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4416256366.pdf","grobid_xml":"https://content.openalex.org/works/W4416256366.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W1971851318","https://openalex.org/W2082490084","https://openalex.org/W4205990911","https://openalex.org/W2026901899","https://openalex.org/W2051579184","https://openalex.org/W2064115678","https://openalex.org/W189016324","https://openalex.org/W2001788313","https://openalex.org/W1639634142","https://openalex.org/W2041043343"],"abstract_inverted_index":{"We":[0],"investigate":[1],"the":[2,9,36,48,59,80,96,110],"convergence":[3,42],"of":[4,8,38,56,82],"signed":[5],"null":[6],"sequences":[7,40,84],"form":[10],"\\[":[11],"\\sum_{n=1}^\\infty":[12],"\\varepsilon_n":[13,16],"a_n,":[14],"\\quad":[15],"\\in":[17],"\\{-1,1\\},":[18],"\\]":[19],"where":[20],"$(a_n)$":[21],"tends":[22],"to":[23],"zero":[24],"in":[25,47,58,73,102],"$\\mathbb{R}^d$.":[26],"Our":[27],"main":[28],"result":[29],"shows":[30],"that":[31],"for":[32,62,77],"any":[33],"such":[34],"sequence,":[35],"set":[37,81],"sign":[39,83],"yielding":[41],"has":[43,89,112],"full":[44,103],"Hausdorff":[45,90],"dimension":[46,91],"natural":[49],"ultrametric":[50],"topology.":[51],"This":[52],"answers":[53],"a":[54],"question":[55],"Mattila":[57],"one-dimensional":[60],"case,":[61],"which":[63],"we":[64],"provide":[65],"an":[66],"elementary":[67],"proof.":[68],"Moreover,":[69],"if":[70,109],"$(a_n)\\notin":[71],"\\ell^1$":[72],"one":[74],"dimension,":[75],"then":[76],"every":[78],"$L\\in\\mathbb{R}$":[79],"with":[85],"sum":[86],"$L$":[87],"also":[88],"$1$.":[92],"In":[93],"higher":[94],"dimensions":[95],"analogous":[97],"statement":[98],"does":[99],"not":[100],"hold":[101],"generality,":[104],"but":[105],"it":[106],"is":[107],"guaranteed":[108],"sequence":[111],"$d$":[113],"linearly":[114],"independent":[115],"L\u00e9vy":[116],"vectors.":[117]},"counts_by_year":[],"updated_date":"2026-11-05T12:25:35.769452","created_date":"2025-10-10T00:00:00"}
