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Dylan Airey

Publications and source records attributed to Dylan Airey.

11 recordsLinked to original sources
arXiv2021

A topological dynamical system with two different positive sofic entropies

A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. All previously known examples showing this dependence rely on degenerate behavior. This paper exhibits an explicit example of a mixing subshift of finite type with two different positive sofic entropies. The example is inspired by statistical physics literature on 2-colorings of random hyper-graphs.

math.DS↗
arXiv2020

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

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 $β$-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 $Π^0_3$-complete set, meaning that it is a countable intersection of $F_σ$-sets, but it is not possible to write it as a countable union of $G_δ$-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 $Π^0_3$-complete.

math.DS↗
arXiv2019

Hotspot lemmas for non-compact spaces

We extend the hotspot lemma, or Piatetskiĭ-Shapiro normality criterion, to dynamical systems on non-compact spaces. We build on the work of N. G. Moshchevitin and I. D. Shkredov, noting an issue in some of their proofs, and adding a necessary and sufficient condition to remove this issue.

math.DS↗
arXiv2016

Some complexity results in the theory of normal numbers

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

math.LO↗
arXiv2016

Normality of different orders for Cantor series expansions

Let $S \subseteq \mathbb{N}$ have the property that for each $k \in S$ the set $(S - k) \cap \mathbb{N} \setminus S$ has asymptotic density $0$. We prove that there exists a basic sequence $Q$ where the set of numbers $Q$-normal of all orders in $S$ but not $Q$-normal of all orders not in $S$ has full Hausdorff dimension. If the function $k \mapsto 1_S(k)$ is computable, then there exist computable examples. For example, there exists a computable basic sequence $Q$ where the set of numbers normal of all even orders and not normal of all odd orders has full Hausdorff dimension. This is in strong constrast to the $b$-ary expansions where any real number that is normal of order $k$ must also be normal of all orders between $1$ and $k-1$. Additionally, all numbers we construct satisfy the unusual condition that block frequencies sampled along non-trivial arithmetic progressions don't converge to the expected value. This is also in strong contrast to the case of the $b$-ary expansions, but more similar to the case of the continued fraction expansion. As a corollary, the set of $Q$-normal numbers that are not normal when sampled along any non-trivial arithmetic progression has full Hausdorff dimension.

math.NT↗
arXiv2014

Normal number constructions for Cantor series with slowly growing bases

Let $Q=(q_n)_{n=1}^\infty$ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both $Q$-normal and $Q$-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$, and from this construction we can provide computable constructions of numbers with atypical normality properties.

math.NT↗
arXiv2014

Normal equivalencies for eventually periodic basic sequences

W. M. Schmidt, A. D. Pollington, and F. Schweiger have studied when normality with respect to one expansion is equivalent to normality with respect to another expansion. Following in their footsteps, we show that when $Q$ is an eventually periodic basic sequence, that $Q$-normality and $Q$-distribution normality are equivalent to normality in base $b$ where $b$ is dependent on $Q$. We also show that boundedness of the basic sequence is not sufficient for this equivalence.

math.NT↗
arXiv2014

Normality preserving operations for Cantor series expansions and associated fractals part II

We investigate how non-zero rational multiplication and rational addition affect normality with respect to $Q$-Cantor series expansions. In particular, we show that there exists a $Q$ such that the set of real numbers which are $Q$-normal but not $Q$-distribution normal, and which still have this property when multiplied and added by rational numbers has full Hausdorff dimension. Moreover, we give such a number that is explicit in the sense that it is computable.

math.NT↗
arXiv2014

Normality preserving operations for Cantor series expansions and associated fractals part I

It is well known that rational multiplication preserves normality in base $b$. We study related normality preserving operations for the $Q$-Cantor series expansions. In particular, we show that while integer multiplication preserves $Q$-distribution normality, it fails to preserve $Q$-normality in a particularly strong manner. We also show that $Q$-distribution normality is not preserved by non-integer rational multiplication on a set of zero measure and full Hausdorff dimension.

math.NT↗