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 to the following pattern. Let $I\in \{M,N,M\cap N,E\}$ and T be a perfect, uniformly perfect or Silver tree. Then for every $A\in I$ there exists $T^{\prime }\subseteq T$ of the same kind as T such that $A+\underset{n\text{–times}}{\underbrace{\left[T^{\prime }\right]+\left[T^{\prime }\right]+\dots +\left[T^{\prime }\right]}}\in I$ for each $n\in \omega$. We also prove weaker statements for splitting trees. For the case $E$ we also provide a simple characterization of a basis of $E$. 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 $c$ is a regular cardinal.