A characteristic factor for the 3-term IP Roth theorem in ℤℕ

Abstract

Let Ω= ⊕∞i=1;ℤ and ei = (0,...,0,1,0,...) where the 1 occurs in the i-th coordinate. There is a natural inclusion of F = {α ⊂ ℕ: ≠αα is finite}. There is a natural inclusion of F into Ω where α ∈ F is mapped to ealpha; = ∑i∈αei. 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.

Publication Title

Electronic Journal of Combinatorics

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