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 $T'\subseteq T$ 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.