On algebraic sums, trees and ideals in the Cantor space

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 $T^{\prime }\subseteq T$ of the same kind as $T$ such that $A+\underset{\text{n–times}}{\underbrace{[T^{\prime }]+[T^{\prime }]+\cdots +[T^{\prime }]}}\in \mathcal{I}$ 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 Sierpiński set belongs to $u_0$ and $v_0$, provided that $\mathfrak{c}$ is a regular cardinal.