Invariant uniformization
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908499558006784 |
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| author | Kechris, Alexander S. Wolman, Michael |
| author_facet | Kechris, Alexander S. Wolman, Michael |
| contents | 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 $\boldsymbol{Σ^1_2}$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_15111 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariant uniformization Kechris, Alexander S. Wolman, Michael Logic 03E15, 28A05, 54H05 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 $\boldsymbol{Σ^1_2}$-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. |
| title | Invariant uniformization |
| topic | Logic 03E15, 28A05, 54H05 |
| url | https://arxiv.org/abs/2405.15111 |