Borel cardinalities below

The Borel cardinality of the quotient of the power set of the natural numbers by the ideal of asymptotically zero-density sets is shown to be the same as that of the equivalence relation induced by the classical Banach space . We also show that a large collection of ideals introduced by Louveau and Velickovic, with pairwise incomparable Borel cardinality, are all Borel reducible to . This refutes a conjecture of Hjorth and has facilitated further work by Farah.