I have been reading about Borel equivalence relations and I have noticed that while $\bf\Sigma^0_3$ equivalence relations are mentioned, there is a conspicuous absence of natural examples (other than $F_{\sigma}$ or $G_{\delta}$ ones).

For example, in Gao's Invariant Descriptive Set Theory he provides examples of $\bf\Pi^0_3$ equivalence relations and even proves that one, $E_0^{\omega}$, is not reducible to any $\bf\Sigma_3^0$ equivalence relation. But he provides no examples of these.

In Kanovei's Borel Equivalence Relations, he proves that all equivalence relations with $\bf\Sigma^0_3$ classes are pinned (exactly what this means isn't important). But again, there are no examples mentioned except those of lower complexity.

Of course examples can be constructed. If $X$ is Polish and $A\subset X$ is $\bf\Sigma_3^0$, you can look at the equivalence relation on $X\times 2$ relating vertical sections with first coordinate in $A$. But this has a Borel selector, so it is smooth.

So, are there natural examples of $\bf\Sigma^0_3$ equivalence relations? Or do all such equivalence relations have to be simple, like how all $G_{\delta}$ equivalence relations are smooth? Or something else? I would be curious to hear about anything that can be said concerning these ERs.


Turing equivalence of real numbers.

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Here is a theorem of Hjorth, Kechris and Louveau which might be of some interest.

Theorem: Let $E^X_G$ be the orbit equivalence relation induced by a Borel action of a closed subgroup $G$ of $Sym(\mathbb{N})$ on some standard Borel space $X$. Then the following are equivalent:

i. $E^X_G$ is essentially countable.

ii. For some Polish topology $\tau$ on $X$ giving its Borel structure, $E^X_G$ is $\mathbf{\Sigma^0_2}$ in $(X,\tau)^2$.

iii. For some Polish topology $\tau$ on $X$ giving its Borel structure, $E^X_G$ is $\mathbf{\Sigma^0_3}$ in $(X,\tau)^2$.

(This is Theorem 1.12 in Jackson, Kechris and Louveau's paper "Countable Borel equivalence relations" and the following papers are given as references for this theorem: Borel equivalence relations induced by actions of the symmetric group, Borel equivalence relations and classifications of countable models. Or, you can check out Chapter 12.5 of Gao's book Invariant Descriptive Set Theory.)

This theorem provides plenty of examples since the isomorphism relation on the standard Borel space of countable $L$-structures is given by the logic action of $Sym(\mathbb{N})$, where $L$ is a countable relational language (see Chapter 3.6 of Gao's book). For example, the isomorphism relation on finitely generated groups is essentially countable.

Of course, even though you might consider these examples "natural", you should keep in mind that we are actually changing the topology without changing the Borel structure, so the topologies which make these equivalence relations $\mathbf{\Sigma^0_3}$ might not be "natural".

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