Borel Cardinalities below C 0

The Borel cardinality of the quotient of the power set of the natural numbers by the ideal Z0 of asymptotically zero-density sets is shown to be the same as that of the equivalence relation induced by the classical Banach space c0. We also show that a large collection of ideals introduced by Louveau and Veličkovič, with pairwise incomparable Borel cardinality, are all Borel reducible to c0. This refutes a conjecture of Hjorth and has facilitated further work by Farah. 1. The ideal of density is equireducible with c0 1.1. Origin of the question. When investigating whether Borel reductions (see Definition 1.3 below) exist between given equivalence relations, it is sometimes convenient to replace one equivalence relation with a combinatorially simpler one which is known to be reducible in both directions with the given equivalence relation. For example, consider the equivalence relation on R induced by the action of 1 by coordinatewise addition (let us refer to this equivalence relation simply as ). Hjorth [Hjo00] has shown that if E ≤B , then either 1 ≤B E, or E is reducible to an equivalence relation all of whose equivalence classes are countable. In his exposition, he replaces 1 with the equivalence relation given by the summable ideal I1/n on the power set of ω: If A ⊆ ω, then A ∈ I1/n just in case ∑ n∈A 1/(n+ 1) < ∞. Since 1 ≤B I1/n and I1/n ≤B , this substitution is legitimate. Here we use the following. Convention 1.1. We write I1/n for the equivalence relation on P(ω) given by A ∼ B ↔ A B ∈ I1/n. Similarly we write Z0 for the equivalence relation on P(ω) induced by Z0 (see Definition 1.2), and c0 and 1 for the equivalence relations on R induced by the actions of those groups by coordinatewise addition. Kechris had suggested that, as 1 is mutually Borel reducible with I1/n, c0 might similarly be equivalent to the ideal of density Z0: Definition 1.2. For A ⊆ ω, A ∈ Z0 ⇐⇒ |A ∩ n| n → 0 as n → ω. Received by the editors April 5, 2004 and, in revised form, October 25, 2004 and February 18, 2005. 2000 Mathematics Subject Classification. Primary 03E15; Secondary 37A20.