A weak dichotomy below E_1 \times E_3

TL;DR

This paper establishes a dichotomy for certain Borel equivalence relations, showing they are either reducible to countable set equality or to E_1, with a strengthened version of the dichotomy.

Contribution

It proves a new dichotomy theorem for equivalence relations reducible to E_1 imes E_3, extending the understanding of their classification.

Findings

If E reduces to E_1 imes E_3, then E is either reducible to countable set equality or E_1 reduces to E.

The dichotomy can be strengthened further in the case where E is reducible to E_1 imes E_3.

The result advances the classification theory of Borel equivalence relations.

Abstract

If E is an equivalence relation Borel reducible to E_1 \times E_3 then either E is Borel reducible to the equality of countable sets of reals or E_1 is Borel reducible to E. The "either" case admits further strengthening.