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Some complexity results in the theory of normal numbers

Part of: Set theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms Number theory: Connections with logic

Published online by Cambridge University Press:  28 September 2020

Dylan Airey ,
Steve Jackson  and
Bill Mance
Dylan Airey
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ08544-1000, USAe-mail:dairey@math.princeton.edu
Steve Jackson*
Affiliation:
Department of Mathematics, University of North Texas, General Academics Building 435, 1155 Union Circle, Denton, #311430, TX76203-5017, USA
Bill Mance
Affiliation:
Uniwersytet im. Adama Mickiewicza w Poznaniu, Collegium Mathematicum, ul. Umultowska 87, Poznań61-614, Polande-mail:william.mance@amu.edu.pl
*
e-mail: stephen.jackson@unt.edu
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Abstract

Let $\mathcal {N}(b)$ be the set of real numbers that are normal to base b. A well-known result of Ki and Linton [19] is that $\mathcal {N}(b)$ is $\boldsymbol {\Pi }^0_3$ -complete. We show that the set ${\mathcal {N}}^\perp (b)$ of reals, which preserve $\mathcal {N}(b)$ under addition, is also $\boldsymbol {\Pi }^0_3$ -complete. We use the characterization of ${\mathcal {N}}^\perp (b),$ given by Rauzy, in terms of an entropy-like quantity called the noise. It follows from our results that no further characterization theorems could result in a still better bound on the complexity of ${\mathcal {N}}^\perp (b)$ . We compute the exact descriptive complexity of other naturally occurring sets associated with noise. One of these is complete at the $\boldsymbol {\Pi }^0_4$ level. Finally, we get upper and lower bounds on the Hausdorff dimension of the level sets associated with the noise.

Keywords

normal numbersnoisecomplexity

MSC classification

Primary: 03E15: Descriptive set theory 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Secondary: 11U99: None of the above, but in this section
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 1 , February 2022 , pp. 170 - 198
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The first author was supported by the National Science Foundation Graduate Research Fellowship grant DGE-1656466. The second author was supported by NSF grant DMS-1800323. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The authors wish to thank the referee for several suggestions which significantly improved the paper.

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