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Transactions of the American Mathematical Society

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Borel complexity of sets of normal numbers via generic points in subshifts with specification


Authors: Dylan Airey, Steve Jackson, Dominik Kwietniak and Bill Mance
Journal: Trans. Amer. Math. Soc. 373 (2020), 4561-4584
MSC (2010): Primary 03E15, 11K16; Secondary 11U99
DOI: https://doi.org/10.1090/tran/8001
Published electronically: April 28, 2020
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Abstract: We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base $ r$ expansions, and their various generalisations: generalised Lüroth series expansions and $ \beta $-expansions. In fact, we consider subshifts over a countable alphabet generated by all possible expansions of numbers in $ [0,1)$. Then normal numbers correspond to generic points of shift-invariant measures. It turns out that for these subshifts the set of generic points for a shift-invariant probability measure is precisely at the third level of the Borel hierarchy (it is a $ \boldsymbol {\Pi }^0_3$-complete set, meaning that it is a countable intersection of $ F_\sigma $-sets, but it is not possible to write it as a countable union of $ G_\delta $-sets). We also solve a problem of Sharkovsky-Sivak on the Borel complexity of the basin of statistical attraction. The crucial dynamical feature we need is a feeble form of specification. All expansions named above generate subshifts with this property. Hence the sets of normal numbers under consideration are $ \boldsymbol {\Pi }^0_3$-complete.


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Additional Information

Dylan Airey
Affiliation: Department of Mathematics, University of Texas at Austin, 2515 Speedway, Austin, Texas 78712-1202
Address at time of publication: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544-1000
Email: dairey@math.princeton.edu

Steve Jackson
Affiliation: Department of Mathematics, University of North Texas, General Academics Building 435, 1155 Union Circle, #311430, Denton, Texas 76203-5017
Email: stephen.jackson@unt.edu

Dominik Kwietniak
Affiliation: Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Łojasiewicza 6, 30-348 Kraków, Poland; and Institute of Mathematics, Federal University of Rio de Janeiro, Cidade Universitaria - Ilha do Fundão, Rio de Janeiro 21945-909, Brazil
Email: dominik.kwietniak@uj.edu.pl

Bill Mance
Affiliation: Institute of Mathematics of Polish Academy of Science, Śniadeckich 8, 00-656 Warsaw, Poland
Address at time of publication: Uniwersytet im. Adama Mickiewicza w Poznaniu, Collegium Mathematicum, ul. Umultowska 87, 61-614 Poznań, Poland
Email: william.mance@amu.edu.pl

DOI: https://doi.org/10.1090/tran/8001
Received by editor(s): October 22, 2018
Received by editor(s) in revised form: July 31, 2019
Published electronically: April 28, 2020
Additional Notes: The second author was supported by NSF grant 1800323.
The third author was supported by National Science Centre (NCN) grant 2013/08/A/ST1/00275 and his stay in Rio de Janeiro, where he started to work on these problems was supported by CAPES/Brazil grant no. 88881.064927/2014-01.
Article copyright: © Copyright 2020 American Mathematical Society