On algebraic sums, trees and ideals in the Baire space

TL;DR

This paper investigates how algebraic sums of certain perfect trees in the Baire space relate to sigma-ideals, providing combinatorial characterizations and extending classical results to this setting.

Contribution

It introduces new results on algebraic sums of perfect trees in the Baire space and characterizes related sigma-ideals using combinatorial methods.

Findings

Established conditions for sums of perfect trees to belong to specific ideals

Extended classical tree and ideal results from Cantor to Baire space

Provided combinatorial characterizations of ideals in the Baire space

Abstract

We work in the Baire space $\mathbb{Z}^\omega$ equipped with the coordinate-wise addition $+$. Consider a $\sigma-$ideal $\mathcal{I}$ and a family $\mathbb{T}$ of some kind of perfect trees. We are interested in results of the form: for every $A\in \mathcal{I}$ and a tree $T\in\mathbb{T}$ there exists $T'\in \mathbb{T}, T'\subseteq T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\in\omega$. Explored tree types include perfect trees, uniformly perfect trees, Miller trees, Laver trees and $\omega-$Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard $\sigma$-ideal $\mathcal{M}$ of meager sets, we also analyze $\mathcal{M}_-$ and fake null sets $\mathcal{N}$. The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space.