arXiv subjects

Steve Jackson

Publications and source records attributed to Steve Jackson.

18 recordsLinked to original sources

arXiv2026

On regulated partitions

This paper considers the combinatorics of continuous and Borel rectangular partitions of free actions of $\mathbb{Z}^n$ on $0$-dimensional Polish spaces, specifically the free part $F(2^{\mathbb{Z}^n})$ of the shift action of $\mathbb{Z}^n$ on the space $2^{\mathbb{Z}^n}$. This is done through the study of a corresponding notion of regulated partitions of $\mathbb{R}^n$. The main concepts studied are the continuous and Borel {\em regulation} numbers of the partition. This is defined as the maximum number of rectangles in the corresponding regulated partition that can intersect in a point. The continuous and Borel regulation numbers $\gamma_c$, $\gamma_B$ are the minimum possible values of these numbers as we range over continuous (respectively Borel) rectangular partitions of $F(2^{\mathbb{Z}^n})$. It is shown that for $n=2$ that $\gamma_c=\gamma_B=3$, and for $n \geq 3$ that $n+2\leq \gamma_B \leq \gamma_c \leq 3\cdot 2^{n-2}$. For $n=3$ we improve this to $\gamma_c=\gamma_B=5$. This shows a striking difference between the Borel combinatorics of dimension $n=2$ and dimensions $n>2$.

math.LO↗

arXiv2024

Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem

We show that several sets of interest arising from the study of partition regularity and density Ramsey theory of polynomial equations over integral domains are undecidable. In particular, we show that the set of homogeneous polynomials $p \in \mathbb{Z}[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{Z}\setminus\{0\}$ is undecidable conditional on Hilbert's tenth problem for $\mathbb{Q}$. For other integral domains, we get the analogous result unconditionally. More generally, we determine the exact lightface complexity of the various sets of interest. For example, we show that the set of homogeneous polynomials $p \in \mathbb{F}_q(t)[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{F}_q(t)\setminus\{0\}$ is $\Pi_2^0$-complete. We also prove several other results of independent interest. These include a compactness principle and a uniformity principle for density Ramsey theory on countable cancellative left amenable semigroups, as well as the existence of the natural extension for measure preserving systems of countable cancellative left reversible semigroups.

math.LO↗

arXiv2024

Borel Combinatorics of Abelian Group Actions

We study the free part of the Bernoulli action of $\mathbb{Z}^n$ for $n\geq 2$ and the Borel combinatorics of the associated Schreier graphs. We construct orthogonal decompositions of the spaces into marker sets with various additional properties. In general, for Borel graphs $\Gamma$ admitting weakly orthogonal decompositions, we show that $\chi_B(\Gamma)\leq 2\chi(\Gamma)-1$ under some mild assumptions. As a consequence, we deduce that the Borel chromatic number for $F(2^{\mathbb{Z}^n})$ is $3$ for all $n\geq 2$. Weakly orthogonal decompositions also give rise to Borel unlayered toast structures. We also construct orthogonal decompositions of $F(2^{\mathbb{Z}^2})$ with strong topological regularity, in particular with all atoms homeomorphic to a disk. This allows us to show that there is a Borel perfect matching for $F(2^{\mathbb{Z}^n})$ for all $n\geq 2$ and that there is a Borel lining of $F(2^{\mathbb{Z}^2})$.

math.LO↗

arXiv2022

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

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, $\alpha$-Birkhoff regular sets and the irregular set appearing in multifractal analysis of ergodic averages of a continuous real-valued 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 $\Pi^0_3$-complete.

math.DS↗

arXiv2021

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

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 $b\ge 2$ is $D_2(\boldsymbol{\Pi}_3^0)$-complete. Moreover, the set of numbers that are normal with respect to the continued fraction expansion but not normal to \emph{any} base-$b$ expansion is $D_2(\boldsymbol{\Pi}_3^0)$-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 $D_2(\boldsymbol{\Pi}_3^0)$-complete. We also prove an auxiliary result on the normality properties of the continued fraction expansions of fractions with a fixed denominator.

math.NT↗

arXiv2020

A non-Borel special alpha-limit set in the square

We consider the complexity of special $\alpha$-limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.

math.DS↗

arXiv2020

Descriptive complexity in Cantor series

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 representation of the form $x=a_0 + \sum_{i=1}^\infty \frac{a_i}{q_1q_2\cdots q_i}$ where $0 \leq a_i<q_i$. These generalize ordinary base $b$ expansions where $q_i=b$. Ki and Linton 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). We show that for any $Q$ the set $\mathscr{DN}(Q)$ of distribution normal number is $\boldsymbol{\Pi}^0_3$-complete, and if $Q$ is $1$-divergent (i.e., $\sum_{i=1}^\infty \frac{1}{q_i}$ diverges) then the sets $\mathscr{N}(Q)$ and $\mathscr{RN}(Q)$ 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 (the trivial case is $\mathscr{N}(Q)\setminus \mathscr{RN}(Q)=\emptyset$). This shows that except for the containment $\mathscr{N}(Q)\subseteq \mathscr{RN}(Q)$, these three notions are as independent as possible.

math.LO↗

arXiv2020

Borel asymptotic dimension and hyperfinite equivalence relations

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of groups including the lamplighter group and all virtually solvable groups having finite Pr\"ufer rank. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov's notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a zero-dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Folner tilings, topological dynamics, and $C^*$-algebras.

math.DS↗

arXiv2018

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 L\"uroth 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 $\Pi^0_3$-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 $\Pi^0_3$-complete.

math.DS↗

arXiv2018

Continuous Combinatorics of Abelian Group Actions

This paper develops techniques which are used to answer a number of questions in the theory of equivalence relations generated by continuous actions of abelian groups. The methods center around the construction of certain specialized hyper-aperiodic elements, which produce compact subflows with useful properties. For example, we show that there is no continuous $3$-coloring of the Cayley graph on $F(2^{\mathbb{Z}^2})$, the free part of the shift action of $\mathbb{Z}^2$ on $2^{\mathbb{Z}^2}$. With earlier work of the authors this computes the continuous chromatic number of $F(2^{\mathbb{Z}^2})$ to be exactly $4$. Combined with marker arguments for the positive directions, our methods allow us to analyze continuous homomorphisms into graphs, and more generally equivariant maps into subshifts of finite type. We present a general construction of a finite set of "tiles" for $2^{\mathbb{Z}^n}$ (there are $12$ for $n=2$) such that questions about the existence of continuous homomorphisms into various structures reduce to finitary combinatorial questions about the tiles. This tile analysis is used to deduce a number of results about $F(2^{\mathbb{Z}^n})$.

math.LO↗

arXiv2017

Folner tilings for actions of amenable groups

We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes") with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-stable.

math.DS↗

arXiv2016

Hyperfiniteness and Borel combinatorics

We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel chromatic number and measure chromatic number. We also show that for every $d > 1$ there is a $d$-regular acyclic hyperfinite Borel bipartite graph with no Borel perfect matching. These techniques also give examples of hyperfinite bounded degree Borel graphs for which the Borel local lemma fails, in contrast to the recent results of Cs\'oka, Grabowski, M\'ath\'e, Pikhurko, and Tyros. Related to the Borel Ruziewicz problem, we show there is a continuous paradoxical action of $(\mathbb{Z}/2\mathbb{Z})^{*3}$ on a Polish space that admits a finitely additive invariant Borel probability measure, but admits no countably additive invariant Borel probability measure. In the context of studying ultrafilters on the quotient space of equivalence relations under $\mathrm{AD}$, we also construct an ultrafilter $U$ on the quotient of $E_0$ which has surprising complexity. In particular, Martin's measure is Rudin-Kiesler reducible to $U$. We end with a problem about whether every hyperfinite bounded degree Borel graph has a witness to its hyperfiniteness which is uniformly bounded below in size.

math.LO↗

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 $\boldsymbol{\Pi}^0_3$-complete. We show that the set $\mathscr{N}(b)$ of reals which preserve $\mathscr{N}(b)$ under addition is also $\boldsymbol{\Pi}^0_3$-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 $\boldsymbol{\Pi}^0_4$ level. Finally, we get upper and lower bounds on the Hausdorff dimension of the level sets associated with the noise.

math.LO↗

arXiv2015

Forcing constructions and countable Borel equivalence relations

We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain features.

math.LO↗

arXiv2013

Determinacy and J\'onsson cardinals in $L(\mathbb{R})$

Assume $\mathsf{ZF}+\mathsf{AD}+V=L(\mathbb{R})$ and let $\kappa<\Theta$ be an uncountable cardinal. We show that $\kappa$ is J\'onsson, and that if $\mathrm{cof}(\kappa)=\omega$ then $\kappa$ is Rowbottom. We also establish some other partition properties.

math.LO↗

arXiv2013

CH, V=L, Disintegrations of Measures, and Π^1_1 Sets

In 1950 Maharam asked whether every disintegration of a $σ$-finite measure into $σ$-finite measures is necessarily uniformly $σ$-finite. Over the years under special conditions on the disintegration, the answer was shown to be yes. However, we show here that the answer may depend on the axioms of set theory in the following sense. If CH, the continuum hypothesis holds, then the answer is no. One proof of this leads to some interesting problems in infinitary combinatorics. If Gödel's axiom of constructibility $\mathbf{V}=\mathbf{L}$ holds, then not only is the answer no, but, of equal interest is the construction of $\mathbfΠ^1_1$ sets with very special properties.

math.LO↗

arXiv2012

Group Colorings and Bernoulli Subflows

In this paper we study the dynamics of Bernoulli flows and their subflows over general countable groups from the symbolic and topological perspectives. We study free subflows (subflows in which every point has trivial stabilizer), minimal subflows, disjointness of subflows, the problem of classifying subflows up to topological conjugacy, and the differences in dynamical behavior between pairs of points which disagree on finitely many coordinates. We call a point hyper aperiodic if the closure of its orbit is a free subflow and we call it minimal if the closure of its orbit is a minimal subflow. We prove that the set of all (minimal) hyper aperiodic points is always dense but also meager and null. By employing notions and ideas from descriptive set theory, we study the complexity of the sets of hyper aperiodic points and of minimal points and completely determine their descriptive complexity. In doing this we introduce a new notion of countable flecc groups and study their properties. We obtain a dichotomy for the complexity of classifying free subflows up to topological conjugacy. For locally finite groups the topological conjugacy relation for all (free) subflows is hyperfinite and nonsmooth. For nonlocally finite groups the relation is Borel bireducible with the universal countable Borel equivalence relation. A primary focus of the paper is to develop constructive methods for the notions studied. To construct hyper aperiodic points, a fundamental method of construction of multi-layer marker structures is developed with great generality. Variations of the fundamental method are used in many proofs in the paper, and we expect them to be useful more broadly in geometric group theory. As a special case of such marker structures, we study the notion of ccc groups and prove the ccc-ness for countable nilpotent, polycyclic, residually finite, locally finite groups and for free products.

math.DS↗

arXiv2009

On the complexity of the uniform homeomorphism relation between separable Banach spaces

We consider the problem of determining the complexity of the uniform homeomorphism relation between separable Banach spaces in the Borel reducibility hierarchy of analytic equivalence relations. We prove that the complete $K_σ$ equivalence relation is Borel reducible to the uniform homeomorphism relation, and we also determine the possible complexities of the relation when restricted to some small classes of Banach spaces. Moreover, we determine the exact complexity of the local equivalence relation between Banach spaces, namely that it is bireducible with $K_σ$. Finally, we construct a class of mutually uniformly homeomorphic Banach spaces such that the equality relation of countable sets of real numbers is Borel reducible to the isomorphism relation on the class.

math.FA↗