RECORD
Invariant uniformization
Logic · Submitted 24 May 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For a given Borel equivalence relation $E$, we show that every $E$-invariant Borel set $P$ with "small" or "large" sections admits an $E$-invariant Borel uniformization if and only if $E$ is smooth.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Standard results in descriptive set theory provide sufficient conditions for a Borel set $P \subseteq \mathbb{N}^\mathbb{N} \times \mathbb{N}^\mathbb{N}$ to admit a Borel uniformization, namely, when $P$ has "small" sections or "large" sections. We consider an invariant analogue of these results: Given a Borel equivalence relation $E$ and an $E$-invariant Borel set $P$ with "small" or "large" sections, does $P$ admit an $E$-invariant Borel uniformization? For a given Borel equivalence relation $E$, we show that every $E$-invariant Borel set $P$ with "small" or "large" sections admits an $E$-invariant Borel uniformization if and only if $E$ is smooth. We also compute the definable complexity of counterexamples in the case where $E$ is not smooth, using category, measure, and Ramsey-theoretic methods. We provide two new proofs of a dichotomy of Miller classifying the pairs $(E, P)$ such that $P$ admits an $E$-invariant uniformization, for a Borel equivalence relation $E$ and a Borel $E$-invariant set $P$ with countable sections. In the process, we prove an $\aleph_0$-dimensional $(\mathbb{G}_0, \mathbb{H}_0)$ dichotomy, generalizing dichotomies of Miller and Lecomte. We also show that the set of pairs $(E, P)$ such that $P$ has "large" sections and admits an $E$-invariant Borel uniformization is $\boldsymbol{\Sigma^1_2}$-complete; in particular, there is no analog of Miller's dichotomy for $P$ with "large" sections. Finally, we consider a less strict notion of invariant uniformization, where we select a countable nonempty subset of each section instead of a single point.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Standard results in descriptive set theory provide sufficient conditions for a Borel set $P \subseteq \mathbb{N}^\mathbb{N} \times \mathbb{N}^\mathbb{N}$ to admit a Borel uniformization, namely, when $P$ has "small" sections or "large" sections. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s1 | We consider an invariant analogue of these results: Given a Borel equivalence relation $E$ and an $E$-invariant Borel set $P$ with "small" or "large" sections, does $P$ admit an $E$-invariant Borel uniformization? | contribution 0.58 | 0.36 | 0.04 | unclear 0.82 |
| s2 | For a given Borel equivalence relation $E$, we show that every $E$-invariant Borel set $P$ with "small" or "large" sections admits an $E$-invariant Borel uniformization if and only if $E$ is smooth. | result 1.00 | 0.94 | 0.82 | asserted 1.00 |
| s3 | We also compute the definable complexity of counterexamples in the case where $E$ is not smooth, using category, measure, and Ramsey-theoretic methods. | contribution 0.54 | 0.95 | 0.05 | asserted 1.00 |
| s4 | We provide two new proofs of a dichotomy of Miller classifying the pairs $(E, P)$ such that $P$ admits an $E$-invariant uniformization, for a Borel equivalence relation $E$ and a Borel $E$-invariant set $P$ with countable sections. | contribution 0.87 | 0.95 | 0.07 | asserted 1.00 |
| s5 | In the process, we prove an $\aleph_0$-dimensional $(\mathbb{G}_0, \mathbb{H}_0)$ dichotomy, generalizing dichotomies of Miller and Lecomte. | result 0.62 | 0.96 | 0.05 | asserted 1.00 |
| s6 | We also show that the set of pairs $(E, P)$ such that $P$ has "large" sections and admits an $E$-invariant Borel uniformization is $\boldsymbol{\Sigma^1_2}$-complete; in particular, there is no analog of Miller's dichotomy for $P$ with "large" sections. | result 0.98 | 0.96 | 0.11 | asserted 1.00 |
| s7 | Finally, we consider a less strict notion of invariant uniformization, where we select a countable nonempty subset of each section instead of a single point. | contribution 0.91 | 0.88 | 0.03 | asserted 0.60 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Standard results in descriptive set theory provide sufficient conditions for a Borel set $P \subseteq \mathbb{N}^\mathbb{N} \times \mathbb{N}^\mathbb{N}$ to admit a Borel uniformization, namely, when $P$ has "small" sections or "large" sections. We consider an invariant analogue of these results: Given a Borel equivalence relation $E$ and an $E$-invariant Borel set $P$ with "small" or "large" sections, does $P$ admit an $E$-invariant Borel uniformization? For a given Borel equivalence relation $E$, we show that every $E$-invariant Borel set $P$ with "small" or "large" sections admits an $E$-invariant Borel uniformization if and only if $E$ is smooth. We also compute the definable complexity of counterexamples in the case where $E$ is not smooth, using category, measure, and Ramsey-theoretic methods. We provide two new proofs of a dichotomy of Miller classifying the pairs $(E, P)$ such that $P$ admits an $E$-invariant uniformization, for a Borel equivalence relation $E$ and a Borel $E$-invariant set $P$ with countable sections. In the process, we prove an $\aleph_0$-dimensional $(\mathbb{G}_0, \mathbb{H}_0)$ dichotomy, generalizing dichotomies of Miller and Lecomte. We also show that the set of pairs $(E, P)$ such that $P$ has "large" sections and admits an $E$-invariant Borel uniformization is $\boldsymbol{\Sigma^1_2}$-complete; in particular, there is no analog of Miller's dichotomy for $P$ with "large" sections. Finally, we consider a less strict notion of invariant uniformization, where we select a countable nonempty subset of each section instead of a single point.
Human review
How the literature uses it
1 arXiv paper cite this in 1 sentence
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.