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Abstract A Cantor series expansion for a real number x with respect to a basic sequence $$Q=(q_1,q_2,\dots )$$ , where $$q_i \geq 2$$ , is a generalization of the base b expansion to an infinite sequence of bases. Ki and Linton in 1994 showed that for ordinary base b expansions the set of normal numbers is a $$\boldsymbol {\Pi }^0_3$$ -complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio normality, and distribution normality. These notions are equivalent for base b expansions, but not for more general Cantor series expansions. We show that for any basic sequence the set of distribution normal numbers is $$\boldsymbol {\Pi }^0_3$$ -complete, and if Q is $$1$$ -divergent then the sets of normal and ratio normal numbers are $$\boldsymbol {\Pi }^0_3$$ -complete. We further show that all five non-trivial differences of these sets are $$D_2(\boldsymbol {\Pi }^0_3)$$ -complete if $$\lim _i q_i=\infty $$ and Q is $$1$$ -divergent. This shows that except for the trivial containment that every normal number is ratio normal, these three notions are as independent as possible.
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Some complexity results in the theory of normal numbers
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.
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- Award ID(s):
- 1800323
- PAR ID:
- 10350246
- Date Published:
- Journal Name:
- Canadian Journal of Mathematics
- Volume:
- 74
- Issue:
- 1
- ISSN:
- 0008-414X
- Page Range / eLocation ID:
- 170 to 198
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4153/S0008414X20000723
