On algebraic sums, trees and ideals in the Cantor space

TL;DR

This paper investigates algebraic sums in the Cantor space, demonstrating how sums of sets within certain ideals and trees preserve ideal membership, and characterizes bases for the ideal of measure zero sets, with implications for generalized Luzin and Sierpiński sets.

Contribution

It establishes new results on algebraic sums involving trees and ideals in the Cantor space, including preservation properties and basis characterizations, extending understanding of set algebra in this context.

Findings

Sum of a set in an ideal with a tree-based sum remains in the ideal.

Weaker sum properties hold for splitting trees.

Characterization of bases for the measure zero ideal .

Abstract

We work in the Cantor space $2^\omega$. The results of the paper adhere the following pattern. Let $\mathcal{I}\in \{\mathcal{M}, \mathcal{N}, \mathcal{M}\cap \mathcal{N}, \mathcal{E}\}$ and $T$ be a perfect, uniformly perfect or Silver tree. Then for every $A\in \mathcal{I}$ there exists $T'\subseteq T$ of the same kind as $T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\in\omega$. We also prove weaker statements for splitting trees. For the case $\mathcal{E}$ we also provide a simple characterization of basis of $\mathcal{E}$. We use these results to prove that the algebraic sum of a generalized Luzin set and a generalized Sierpi\'nski set belongs to $u_0$ and $v_0$, provided that $\mathfrak{c}$ is a regular cardinal.