{"doi":"10.1017/jsl.2020.73","title":"LUZIN’S (N) AND RANDOMNESS REFLECTION","abstract":"<jats:title>Abstract</jats:title><jats:p>We show that a computable function <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481220000730_inline1.png\" /><jats:tex-math>\n$f:\\mathbb R\\rightarrow \\mathbb R$\n</jats:tex-math></jats:alternatives></jats:inline-formula> has Luzin’s property (N) if and only if it reflects <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481220000730_inline2.png\" /><jats:tex-math>\n$\\Pi ^1_1$\n</jats:tex-math></jats:alternatives></jats:inline-formula>-randomness, if and only if it reflects <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481220000730_inline3.png\" /><jats:tex-math>\n$\\Delta ^1_1({\\mathcal {O}})$\n</jats:tex-math></jats:alternatives></jats:inline-formula>-randomness, and if and only if it reflects <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481220000730_inline4.png\" /><jats:tex-math>\n${\\mathcal {O}}$\n</jats:tex-math></jats:alternatives></jats:inline-formula>-Kurtz randomness, but reflecting Martin–Löf randomness or weak-2-randomness does not suffice. Here a function <jats:italic>f</jats:italic> is said to reflect a randomness notion <jats:italic>R</jats:italic> if whenever <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481220000730_inline5.png\" /><jats:tex-math>\n$f(x)$\n</jats:tex-math></jats:alternatives></jats:inline-formula> is <jats:italic>R</jats:italic>-random, then <jats:italic>x</jats:italic> is <jats:italic>R</jats:italic>-random as well. If additionally <jats:italic>f</jats:italic> is known to have bounded variation, then we show <jats:italic>f</jats:italic> has Luzin’s (N) if and only if it reflects weak-2-randomness, and if and only if it reflects <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481220000730_inline6.png\" /><jats:tex-math>\n$\\emptyset '$\n</jats:tex-math></jats:alternatives></jats:inline-formula>-Kurtz randomness. This links classical real analysis with algorithmic randomness.</jats:p>","journal":"The Journal of Symbolic Logic","year":2022,"id":42303,"datarank":0.10397207708399181,"base_score":0.6931471805599453,"endowment":0.6931471805599453,"self_citation_contribution":0.10397207708399181,"citation_network_contribution":0.0,"self_endowment_contribution":0.10397207708399181,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":1,"citer_count":1,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":null,"is_data_producer":false,"deposit_databanks":null,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":null,"fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":202643,"name":"LINDA WESTRICK","orcid":null,"position":1,"is_corresponding":false},{"id":202641,"name":"LIANG YU","orcid":null,"position":2,"is_corresponding":false},{"id":202642,"name":"ARNO PAULY","orcid":null,"position":0,"is_corresponding":false}],"reference_count":0,"raw_metadata":{"has_enrichment":true,"base_score":0.6931471805599453,"endowment":0.6931471805599453,"datacite_reuse_total":0,"file_count":0,"downloads":0,"views":0,"has_version_chain":false,"is_dataset":false,"is_oa":false,"pmid":"18998881","pmcid":null,"openalex_id":"https://openalex.org/W3094954546","authors":[],"funders":[{"funder_name":"European Commission","grant_id":"731143","title":"Computing with Infinite Data"}],"total_grants":1,"fwci":0.1457,"citation_percentile":0.55333467,"influential_citations":0,"citation_trend":[{"year":2022,"count":1}],"oa_status":"bronze","license":"Cambridge Core User Agreement","oa_locations":[{"url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/0EB472454C7B8525BC3CB09E9F0C31F0/S0022481220000730a.pdf/div-class-title-luzin-s-n-and-randomness-reflection-div.pdf","host_type":"journal"},{"url":"https://cronfa.swan.ac.uk/Record/cronfa57510/Download/57510__20638__1f2aff8b94e941188e9749241bc2aa51.pdf","host_type":"GREEN"},{"url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/0EB472454C7B8525BC3CB09E9F0C31F0/S0022481220000730a.pdf/div-class-title-luzin-s-n-and-randomness-reflection-div.pdf","host_type":"publisher"},{"url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022481220000730","host_type":"publisher"},{"url":"https://doi.org/10.1017/jsl.2020.73","host_type":"journal"},{"url":"https://dx.doi.org/10.48550/arxiv.2006.07517","host_type":""},{"url":"http://arxiv.org/abs/2006.07517","host_type":""},{"url":"https://zbmath.org/7541923","host_type":""},{"url":"https://dx.doi.org/10.1017/jsl.2020.73","host_type":""},{"url":"https://cronfa.swan.ac.uk/Record/cronfa57510","host_type":""},{"url":"http://dx.doi.org/10.1017/jsl.2020.73","host_type":""},{"url":"https://doi.org/https://doi.org/10.1017/jsl.2020.73","host_type":""}],"fields_of_study":["Computability, Logic, AI Algorithms","Benford’s Law and Fraud Detection","Advanced Topology and Set Theory","Mathematics","Computer Science","0101 mathematics","01 natural sciences"],"mesh_terms":[],"keywords":["Randomness","Bounded function","Mathematics","Combinatorics","Function (biology)","Discrete mathematics","Mathematical analysis","Statistics","computable analysis","Algorithmic randomness and dimension","FOS: Mathematics","03D32, 26A30","higher randomness","randomness reflection","Mathematics - Logic","Computation over the reals, computable analysis","Logic (math.LO)","Luzin's (N)"],"sdg_mappings":[],"linked_datasets":[],"clinical_trials":[],"software_tools":[],"database_accessions":[],"source":"live","citation_network_status":"fetched"},"created_at":"2026-06-14T02:12:33.470314Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}