The descriptive complexity of the set of Poisson generic numbers

IF 0.9 1区数学 Q1 LOGIC Journal of Mathematical Logic Pub Date : 2024-05-09 DOI:10.1142/s0219061324500193

Verónica Becher, Stephen Jackson, Dominik Kwietniak, Bill Mance

{"title":"The descriptive complexity of the set of Poisson generic numbers","authors":"Verónica Becher, Stephen Jackson, Dominik Kwietniak, Bill Mance","doi":"10.1142/s0219061324500193","DOIUrl":null,"url":null,"abstract":"Let <math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi><mo>≥</mo><mn>2</mn></math> be an integer. We show that the set of real numbers that are Poisson generic in base <math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi></math> is <math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mstyle><mtext mathvariant=\"normal\">Π</mtext></mstyle></mrow><mrow><mn>3</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math>-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base <math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi></math> and not Poisson generic in base <math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi></math> is complete for the class given by the differences between <math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mstyle><mtext mathvariant=\"normal\">Π</mtext></mstyle></mrow><mrow><mn>3</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math> sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219061324500193","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"LOGIC","Score":null,"Total":0}

引用次数: 0

Abstract

Let $b \geq 2$ be an integer. We show that the set of real numbers that are Poisson generic in base $b$ is $Π_{3}^{0}$ -complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base $b$ and not Poisson generic in base $b$ is complete for the class given by the differences between $Π_{3}^{0}$ sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

泊松通用数集的描述复杂性

设 b≥2 为整数。我们证明，在基 b 中为泊松泛函的实数集合在实线子集的伯尔层次中是 Π30 完全的。此外，对于由 Π30 集之间的差异给出的类来说，在基 b 中是伯尔正则且在基 b 中不是泊松泛函的实数集是完备的。我们还证明了这些结果的有效版本在有效伯尔层次中成立。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Journal of Mathematical Logic MATHEMATICS-LOGIC

CiteScore

1.60

自引率

11.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Mathematical Logic (JML) provides an important forum for the communication of original contributions in all areas of mathematical logic and its applications. It aims at publishing papers at the highest level of mathematical creativity and sophistication. JML intends to represent the most important and innovative developments in the subject.

期刊最新文献

The descriptive complexity of the set of Poisson generic numbers The equivalence of Axiom (*)+ and Axiom (*)++ Non-Galvin filters On the consistency of ZF with an elementary embedding from Vλ+2 into Vλ+2 Rings of finite Morley rank without the canonical base property