arXiv subjects

Stephen Jackson

Publications and source records attributed to Stephen Jackson.

15 recordsLinked to original sources

arXiv2026

Schmidt's Game and Vitali Sets

While many types of non-measurable sets are never $(\alpha, \beta)$-winning in the sense of Schmidt's game, we show that this is not the case for certain Vitali sets. Our main theorems show that for certain values of $\alpha, \beta$ one can construct a Vitali set which is $(\alpha, \beta)$-winning, while for other values of $\alpha,\beta$ every Vitali set is $(\alpha,\beta)$-losing. We also investigate the $(\alpha,\beta)$-Schmidt game for various other types of pathological sets, highlighting their differences from Vitali sets.

math.LO↗

arXiv2025

On coarse geometry of separable dual Banach spaces

We study the obstructions to coarse universality in separable dual Banach spaces. We prove coarse non-universality of several classes of dual spaces, including those with conditional spreading bases, as well as generalized James and James tree spaces. We also give quantitative counterparts of some of the results, clarifying the distinction between coarse non-universality and the non-equi-coarse embeddings of the Kalton graphs. Unique to our approach is the use of a Ramsey ultrafilter. While the existence of such ultrafilters typically requires $\mathsf{CH}$, we are able to show that the conclusions of our theorems follow from $\mathsf{ZFC}$, alone via an absoluteness argument. Finally, we also show how our techniques can be used to prove various previously known results in the literature.

math.FA↗

arXiv2023

The descriptive complexity of the set of Poisson generic numbers

Let $b\ge 2$ be an integer. We show that the set of real numbers that are Poisson generic in base $b$ is $\boldsymbol{\Pi}^0_3$-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base $b$ and not Poisson generic in base $b$ is complete for the class given by the differences between $\boldsymbol{\Pi}^0_3$ sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.

math.LO↗

arXiv2022

Complexity and Ramsey Largeness of Sets of Oracles Separating Complexity Classes

We prove two sets of results concerning computational complexity classes. The first concerns a variation of the random oracle hypothesis posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, P not equal NP with probability 1. This hypothesis was quickly disproven in several ways, most famously in 1992 with the result that IP equals PSPACE, in spite of the classes being shown unequal with probability 1. Here we propose a variation of what it means to be ``large'' using the Ellentuck topology. In this new context, we demonstrate that the set of oracles separating NP and co-NP is not small, and obtain similar results for the separation of PSPACE from PH along with the separation of NP from BQP. We demonstrate that this version of the hypothesis turns it into a sufficient condition for unrelativized relationships, at least in the three cases considered here. Second, we example the descriptive complexity of the classes of oracles providing the separations for these various classes, and determine their exact placement in the Borel hierarchy.

math.LO↗

arXiv2021

The no-$\beta$ McMullen game and the perfect set property

Given a target set $A\subseteq \mathbb{R}^d$ and a real number $\beta\in (0,1)$, McMullen introduced the notion of $A$ being an absolutely $\beta$-winning set. This involves a two player game which we call the $\beta$-McMullen game. We consider the version of this game in which the parameter $\beta$ is removed, which we call the no-$\beta$ McMullen game. More generally, we consider the game with respect to arbitrary norms on $\mathbb{R}^d$, and even more generally with respect to general convex sets. We show that for strictly convex sets in $\mathbb{R}^d$, polytopes in $\mathbb{R}^d$, and general convex sets in $\mathbb{R}^2$, that player $\boldsymbol{I}$ wins the no-$\beta$ McMullen game iff $A$ contains a perfect set and player $\boldsymbol{I}\kern-0.05cm\boldsymbol{I}$ wins iff $A$ is countable. So, the no-$\beta$ McMullen game is equivalent to the perfect set game for $A$ in these cases. The proofs of these results use a connection between the geometry of the game and techniques from logic. Because of the geometry of this game, this result has strong implications for the geometry of uncountable sets in $\mathbb{R}^d$. We also present an example of a compact, convex set in $\mathbb{R}^3$ to which our methods do not apply, and also an example due to D.\ Simmons of a closed, convex set in $\ell_2(\mathbb{R})$ which illustrate the obstacles in extending the results further.

math.LO↗

arXiv2021

On the existence of numbers with matching continued fraction and decimal expansions

A Trott number is a number $x\in(0,1)$ whose continued fraction expansion is equal to its base $b$ expansion for a given base $b$, in the following sense: If $x=[0;a_1,a_2,\dots]$, then $x=(0.\hat{a}_1\hat{a}_2\dots)_b$, where $\hat{a}_i$ is the string of digits resulting from writing $a_i$ in base $b$. In this paper we characterize the set of bases for which Trott numbers exist, and show that for these bases, the set $T_b$ of Trott numbers is a complete $G_\delta$ set. We prove moreover that the union $T:=\bigcup_{b\geq 2} T_b$ is nowhere dense and has Hausdorff dimension less than one. Finally, we give several sufficient conditions on bases $b$ and $b'$ such that $T_b\cap T_{b'}=\emptyset$, and conjecture that this is the case for all $b\neq b'$. This question has connections with some deep theorems in Diophantine approximation.

math.NT↗

arXiv2020

Hausdorff Dimension Regularity Properties and Games

The Hausdorff $\delta$-dimension game was introduced by Das, Fishman, Simmons and {Urba{\'n}ski} and shown to characterize sets in $\mathbb{R}^d$ having Hausdorff dimension $\leq \delta$. We introduce a variation of this game which also characterizes Hausdorff dimension and for which we are able to prove an unfolding result similar to the basic unfolding property for the Banach-Mazur game for category. We use this to derive a number of consequences for Hausdorff dimension. We show that under $\mathsf{AD}$ any wellordered union of sets each of which has Hausdorff dimension $\leq \delta$ has dimension $\leq \delta$. We establish a continuous uniformization result for Hausdorff dimension. The unfolded game also provides a new proof that every $\boldsymbol{\Sigma}^1_1$ set of Hausdorff dimension $\geq \delta$ contains a compact subset of dimension $\geq \delta'$ for any $\delta'<\delta$, and this result generalizes to arbitrary sets under $\mathsf{AD}$.

math.LO↗

arXiv2020

Equivalence Relations and Determinacy

We introduce the notion of $(\Gamma,E)$-determinacy for $\Gamma$ a pointclass and $E$ an equivalence relation on a Polish space $X$. A case of particular interest is the case when $E=E_G$ is the (left) shift-action of $G$ on $S^G$ where $S=2=\{0,1\}$ or $S=\omega$. We show that for all shift actions by countable groups $G$, and any "reasonable" pointclass $\Gamma$, that $(\Gamma,E_G)$-determinacy implies $\Gamma$-determinacy. We also prove a corresponding result when $E$ is a subshift of finite type of the shift map on $2^\mathbb{Z}$.

math.LO↗

arXiv2019

The Measure Game

We study a game first introduced by Martin (actually we use a slight variation of this game) which plays a role for measure analogous to the Banach-Mazur game for category. We first present proofs for the basic connections between this game and measure, and then use the game to prove fundamental measure theoretic results such as Fubini's theorem, the Borel-Cantelli lemma, and a general unfolding result for the game which gives, for example, the measurability of $\boldsymbol{\Sigma}^1_1$ sets. We also use the game to give a new, more constructive, proof of a strong form of the R\'{e}nyi-Lamperti lemma, an important result in probability theory with many applications to number theory. The proofs we give are all direct combinatorial arguments using the game, and do not depend on known measure theoretic arguments.

math.LO↗

arXiv2019

The Destruction of the Axiom of Determinacy by Forcings on $\mathbb{R}$ when $\Theta$ is Regular

$\mathsf{ZF + AD}$ proves that for all nontrivial forcings $\mathbb{P}$ on a wellorderable set of cardinality less than $\Theta$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} \neg\mathsf{AD}$. $\mathsf{ZF + AD} + \Theta$ is regular proves that for all nontrivial forcing $\mathbb{P}$ which is a surjective image of $\mathbb{R}$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} \neg\mathsf{AD}$. In particular, $\mathsf{ZF + AD + V = L(\mathbb{R})}$ proves that for every nontrivial forcing $\mathbb{P} \in L_\Theta(\mathbb{R})$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} \neg\mathsf{AD}$.

math.LO↗

arXiv2019

Cardinality of Wellordered Disjoint Unions of Quotients of Smooth Equivalence Relations

Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. Let $\approx$ denote the relation of being in bijection. Let $\kappa \in \mathrm{ON}$ and $\langle E_\alpha : \alpha < \kappa\rangle$ be a sequence of equivalence relations on $\mathbb{R}$ with all classes countable and for all $\alpha < \kappa$, $\mathbb{R} / E_\alpha \approx \mathbb{R}$. Then the disjoint union $\bigsqcup_{\alpha < \kappa} \mathbb{R} / E_\alpha$ is in bijection with $\mathbb{R} \times \kappa$ and $\bigsqcup_{\alpha < \kappa} \mathbb{R} / E_\alpha$ has the J\'onsson property. Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. A set $X \subseteq [\omega_1]^{<\omega_1}$ has a sequence $\langle E_\alpha : \alpha < \omega_1\rangle$ of equivalence relations on $\mathbb{R}$ such that $\mathbb{R} / E_\alpha \approx \mathbb{R}$ and $X \approx \bigsqcup_{\alpha < \omega_1} \mathbb{R} / E_\alpha$ if and only if $\mathbb{R} \sqcup \omega_1$ injects into $X$. Assume $\mathsf{AD}$. Suppose $R \subseteq [\omega_1]^\omega \times \mathbb{R}$ is a relation such that for all $f \in [\omega_1]^\omega$, $R_f = \{x \in \mathbb{R} : R(f,x)\}$ is nonempty and countable. Then there is an uncountable $X \subseteq \omega_1$ and function $\Phi : [X]^\omega \rightarrow \mathbb{R}$ which uniformizes $R$ on $[X]^\omega$: that is, for all $f \in [X]^\omega$, $R(f,\Phi(f))$. Under $\mathsf{AD}$, if $\kappa$ is an ordinal and $\langle E_\alpha : \alpha < \kappa\rangle$ is a sequence of equivalence relations on $\mathbb{R}$ with all classes countable, then $[\omega_1]^\omega$ does not inject into $\bigsqcup_{\alpha < \kappa} \mathbb{R} / E_\alpha$.

math.LO↗

arXiv2018

$L(\mathbb{R})$ with Determinacy Satisfies the Suslin Hypothesis

The Suslin hypothesis states that there are no nonseparable complete dense linear orderings without endpoints which have the countable chain condition. $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$ proves the Suslin hypothesis. In particular, if $L(\mathbb{R}) \models \mathsf{AD}$, then $L(\mathbb{R})$ satisfies the Suslin hypothesis, which answers a question of Foreman.

math.LO↗

arXiv2018

Equivalence of the Rothberger and $2$-Rothberger Games for Hausdorff Spaces

We prove that in any Hausdorff space, the Rothberger game is equivalent to the $k$-Rothberger game, i.e. the game in which player II chooses $k$ open sets in each move. This result follows from a more general theorem in which we show these games are equivalent to a game we call the restricted Menger game. In this game I knows immediately in advance of playing each open cover how many open sets II will choose from that open cover. This result illuminates the relationship between the Rothberger and Menger games in Hausdorff spaces. The equivalence of these games answers a question posed by Aurichi, Bella, and Dias, at least in the context of Hausdorff spaces.

math.GN↗

arXiv2017

Determinacy of Schmidt's Game and Other Intersection Games

Schmidt's game, and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, $\mathsf{AD}_\mathbb{R}$, which is a much stronger axiom than that asserting all integer games are determined, $\mathsf{AD}$. One of our main results is a general theorem which under the hypothesis $\mathsf{AD}$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt's $(\alpha,\beta,\rho)$ game on $\mathbb{R}$ is determined from $\mathsf{AD}$ alone, but on $\mathbb{R}^n$ for $n \geq 3$ we show that $\mathsf{AD}$ does not imply the determinacy of this game. We also prove several other results specifically related to the determinacy of Schmidt's game. These results highlight the obstacles in obtaining the determinacy of Schmidt's game from $\mathsf{AD}$.

math.LO↗

arXiv2007

On the Structure and Properties of Differentially Rotating Main-Sequence Stars in the 1-2 M_sun Range

We conduct a systematic examination of the properties of models for chemically homogeneous, differentially rotating, main-sequence stars of mass 1-2 M_sun. The models were constructed using a code based on a reformulation of the self-consistent field method of computing the equilibrium stellar structure for a specified conservative internal rotation law. [abridged] Relative to nonrotating stars of the same mass, these models all have reduced luminosities and effective temperatures, and flattened photospheric shapes (i.e., decreased polar radii) with equatorial radii that can be larger or smaller, depending on the degree of differential rotation. For a fixed ratio of the axial rotation rate to the surface equatorial rotation rate, increasingly rapid rotation generally deepens convective envelopes, shrinks convective cores, and can lead to the presence of a convective core (envelope) in a 1 M_sun (2 M_sun) model, a feature that is absent in a nonrotating star of the same mass. The positions of differentially rotating models for a given mass M in the H-R diagram can be shifted in such a way as to approximate the nonrotating ZAMS over ranges in luminosity and effective temperature that correspond to a mass interval between M and about 0.7 M. We briefly note a few of the implications of these results, including (i) possible ambiguities arising from similarities between the properties of rotating and nonrotating models of different masses, (ii) a reduced radiative luminosity for a young, rapidly rotating Sun, (iii) the nuclear destruction of lithium and other light metallic species in the layers beneath an outer convective envelope, and (iv), the excitation of solar-like oscillations and the operation of a solar-like hydromagnetic dynamo in some 1.5-2 M_sun stars.

astro-ph↗