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Benjamin D. Miller

Publications and source records attributed to Benjamin D. Miller.

18 recordsLinked to original sources
arXiv2022

A ratio ergodic theorem via tiling and uniformly syndetic markers

We prove a purely Borel/measureless version of Dowker's ratio ergodic theorem, from which we derive a strengthening of Dowker's original theorem with a precise identification of the limit of local ergodic ratios. This is done by implementing the pointwise tiling idea of [Tse18] in the more complex setting of continuum-to-one Borel transformations. Along the way, we establish a vanishing markers lemma for these transformations, which generalizes its well-known counterpart for invertible transformations.

math.DS↗
arXiv2020

Sigma-continuity with closed witnesses

We use variants of the $\mathbb{G}_0$ dichotomy to establish a refinement of Solecki's basis theorem for the family of Baire-class one functions which are not $σ$-continuous with closed witnesses.

math.LO↗
arXiv2020

Lacunary sets for actions of tsi groups

Under a mild definability assumption, we characterize the family of Borel actions $Γ\curvearrowright X$ of tsi Polish groups on Polish spaces that can be decomposed into countably-many actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of $1_Γ$. In the special case that $Γ$ is non-archimedean, it follows that there is such a decomposition if and only if there is no continuous embedding of $\mathbb{E}_0^{\mathbb{N}}$ into $E_Γ^X$.

math.LO↗
arXiv2020

Bases for functions beyond the first Baire class

We provide a finite basis for the class of Borel functions that are not in the first Baire class, as well as the class of Borel functions that are not $σ$-continuous with closed witnesses.

math.LO↗
arXiv2020

Incomparable actions of free groups

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $μ$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $Γ$, there is a Borel sequence $(\cdot_r)_{r \in \mathbb{R}}$ of free actions of $Γ$ on $X$, generating subequivalence relations $E_r$ of $E$ with respect to which $μ$ is ergodic, with the further property that $(E_r)_{r \in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $μ$-reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on $X$, generating a subequivalence relation of $E$ with respect to which $μ$ is ergodic.

math.LO↗
arXiv2020

Measurable perfect matchings for acyclic locally countable Borel graphs

We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating $μ$-hyperfinite equivalence relations admit $μ$-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a counterexample in the locally countable case.

math.LO↗
arXiv2020

Measure reducibility of countable Borel equivalence relations

We show that every basis for the countable Borel equivalence relations strictly above $\mathbb{E}_0$ under measure reducibility is uncountable, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base, using arguments substantially simpler than those previously employed.

math.LO↗
arXiv2020

Recurrence and the existence of invariant measures

We show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.

math.LO↗
arXiv2019

Minimal definable graphs of definable chromatic number at least three

We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property, and show that the analogous result for digraphs fails.

math.LO↗
arXiv2018

On the existence of large antichains for definable quasi-orders

We generalize Harrington-Marker-Shelah's Dilworth-style characterization of the existence of non-empty perfect antichains to co-analytic quasi-orders, establish the analogous theorem at the next definable cardinal, and consider generalizations beyond the first level of the projective hierarchy.

math.LO↗
arXiv2018

Edge sliding and ergodic hyperfinite decomposition

We use edge slidings and saturated disjoint Borel families to give a conceptually simple proof of Hjorth's theorem on cost attained: if a countable p.m.p. ergodic equivalence relation $E$ is treeable and has cost $n \in \mathbb{N} \cup \{\infty\}$ then it is induced by an a.e. free p.m.p. action of the free group $\mathbb{F}_n$ on $n$ generators. More importantly, our techniques give a significant strengthening of this theorem: the action of $\mathbb{F}_n$ can be arranged so that each of the $n$ generators alone acts ergodically. The existence of an ergodic action for the first generator immediately follows from a powerful theorem of Tucker-Drob, whose proof however uses a recent substantial result in probability theory as a black box. We give a constructive and purely descriptive set theoretic proof of a weaker version of Tucker-Drob's theorem, which is enough for many of its applications, including our strengthening of Hjorth's theorem. Our proof uses new tools, such as asymptotic means on graphs, packed disjoint Borel families, and a cost threshold for finitizing the connected components of nonhyperfinite graphs.

math.DS↗
arXiv2018

The open dihypergraph dichotomy and the second level of the Borel hierarchy

We show that several dichotomy theorems concerning the second level of the Borel hierarchy are special cases of the $\aleph_0$-dimensional generalization of the open graph dichotomy, which itself follows from the usual proof(s) of the perfect set theorem. Under the axiom of determinacy, we obtain the generalizations of these results from analytic metric spaces to separable metric spaces. We also consider connections between cardinal invariants and the chromatic numbers of the corresponding dihypergraphs.

math.LO↗
arXiv2014

Dichotomy Theorems for Families of Non-Cofinal Essential Complexity

We prove that for every Borel equivalence relation $E$, either $E$ is Borel reducible to $\mathbb{E}\_0$, or the family of Borel equivalence relations incompatible with $E$ has cofinal essential complexity. It follows that if $F$ is a Borel equivalence relation and $\cal F$ is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation $E$, either $E\in {\cal F}$ or $F$ is Borel reducible to $E$, then $\cal F$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.

math.LO↗
arXiv2014

Essential countability of treeable equivalence relations

We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of E1 into E. Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one.

math.LO↗