Coanalyticity of I_W for Borel W not in Σ^0_2

Ascertain whether, for every Borel subset W of the Cantor space {0,1}^ω that is not Σ^0_2, the ideal I_W := {A⊆ω : L_{x|A} ∩ W = ∅}, defined using a fixed dense sequence x enumerating the finitely supported sequences in {0,1}^ω (with x_0 the constant zero sequence), is always coanalytic in P(ω). Here L_{x|A} denotes the set of ordinary limit points of the subsequence (x_n) restricted to A.

Background

The paper introduces the ideal I_W associated to a subset W of a topological space X and a sequence x, defined by I_W = {A⊆ω : L_{x|A} ∩ W = ∅}. Specializing to X being the Cantor space and x a dense enumeration of its rationals (finitely supported sequences), the authors analyze the descriptive complexity of I_W.

They prove: if W is closed then I_W is Σ^0_2; if W is Σ^0_2 then I_W is Π^0_3; if W is open then I_W is an analytic P-ideal; and if W is Borel but not Σ^0_2 then I_W is not analytic. They also show I_{W_irr} (with W_irr the set of sequences with infinitely many 1’s) is coanalytic and Π^1_1-complete. The remaining question is whether I_W is always coanalytic for all Borel W not in Σ^0_2.