# A characteristic factor for the 3-term IP Roth theorem in ℤ^{ℕ}

## 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

## Recommended Citation

McCutcheon, R., & Windsor, A.
(2014). A characteristic factor for the 3-term IP Roth theorem in ℤ^{ℕ}.* Electronic Journal of Combinatorics**, 21* (3)
Retrieved from https://digitalcommons.memphis.edu/facpubs/3988