Some complexity results in the theory of normal numbers

Airey, Dylan; Jackson, Steve; Mance, Bill

doi:10.4153/s0008414x20000723

Cited by 4 publications

(11 citation statements)

References 18 publications

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“…2 One can also define D β (Π 0 α ) more generally, but this will not be relevant to this paper. 3 Technically there are two universal quantifiers, one for s and one for ǫ, but as these are consecutive, they are treated as a single quantifier Since any countable set can be written as a countable union of singleton sets and is thus in Σ 0 2 , the above theorem implies the first half of Corollary 1.1 immediately. The second half is implicit in the method of proof.…”

Section: Introductionmentioning

confidence: 96%

On the Borel complexity of continued fraction normal, absolutely abnormal numbers

Jackson¹,

Mance²,

Vandehey³

2021

Preprint

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We show that normality for continued fractions expansions and normality for base-b expansions are maximally logically separate. In particular, the set of numbers that are normal with respect to the continued fraction expansion but not base-b normal for a fixed b ≥ 2 is D 2 (Π 0 3 )-complete. Moreover, the set of numbers that are normal with respect to the continued fraction expansion but not normal to any base-b expansion is D 2 (Π 0 3 )-hard, confirming the existence of uncountably many such numbers, which was previously only known assuming the generalized Riemann hypothesis.By varying the method of proof we are also able to show that the set of base-2 normal, base-3 non-normal numbers is also D 2 (Π 0 3 )-complete. We also prove an auxiliary result on the normality properties of the continued fraction expansions of fractions with a fixed denominator.Date: November 24, 2021. 1 If we ignore rational numbers, then we can guarantee all CF expansions are infinite.

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Section: Introductionmentioning

confidence: 96%

On the Borel complexity of continued fraction normal, absolutely abnormal numbers

Jackson¹,

Mance²,

Vandehey³

2021

Preprint

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“…Furthermore, V. Becher and T. A. Slaman [5] proved that the set of numbers normal in at least one base is Σ 0 4 -complete. See also [1], [6]. Also, sets of interest in dynamics are sometimes complete at levels past the Borel hierarchy.…”

Section: Introductionmentioning

confidence: 99%

Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property

Deka¹,

Jackson²,

Kwietniak³

et al. 2022

Preprint

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We study the descriptive complexity of sets of points defined by placing restrictions on statistical behaviour of their orbits in dynamical systems on Polish spaces. A particular examples of such sets are the set of generic points of a T -invariant Borel probability measure, but we also consider much more general sets (for example, α-Birkhoff regular sets and the irregular set appearing in multifractal analysis of ergodic averages of a continuous realvalued function). We show that many of these sets are Borel. In fact, all these sets are Borel when we assume that our space is compact. We provide examples of these sets being non-Borel, properly placed at the first level of the projective hierarchy (they are complete analytic or co-analytic). This proves that the compactness assumption is in some cases necessary to obtain Borelness. When these sets are Borel, we use the Borel hierarchy to measure their descriptive complexity. We show that the sets of interest are located at most at the third level of the hierarchy. We also use a modified version of the specification property to show that for many dynamical systems these sets are properly located at the third level. To demonstrate that the specification property is a sufficient, but not necessary condition for maximal descriptive complexity of a set of generic points, we provide an example of a compact minimal system with an invariant measure whose set of generic points is Π 0 3complete.

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“…Since the upper noise of is defined by taking limits (and lim sups) of sequences of continuous functions of , it follows immediately that is Borel, in fact a set. In fact, in [3] it was shown that is a -complete set, which shows that it is no simpler than this.…”

Section: Introductionmentioning

confidence: 99%