A Characteristic Factor for the 3-Term IP Roth Theorem in Z3

Let Ω = ⊕∞ i=1 Z3 and ei = (0, . . . , 0, 1, 0, . . . ) where the 1 occurs in the i-th coordinate. Let F = {α ⊂ N : ∅ 6= α, α is finite}. There is a natural inclusion of F into Ω where α ∈ F is mapped to eα = ∑ i∈α ei. We give a new proof that if E ⊂ Ω with d∗(E) > 0 then there exist ω ∈ Ω and α ∈ F such that {ω, ω + eα, ω + 2eα} ⊂ E. Our proof establishes that for the ergodic reformulation of the problem there is a characteristic factor that is a one step compact extension of the Kronecker factor.