Topological and measure properties of some self-similar sets

Given a finite subset $\Sigma\subset\mathbb{R}$ and a positive real number $q<1$ we study topological and measure-theoretic properties of the self-similar set $K(\Sigma;q)=\big\{\sum_{n=0}^\infty a_nq^n:(a_n)_{n\in\omega}\in\Sigma^\omega\big\}$, which is the unique compact solution of the equation $K=\Sigma+qK$. The obtained results are applied to studying partial sumsets $E(x)=\big\{\sum_{n=0}^\infty x_n\varepsilon_n:(\varepsilon_n)_{n\in\omega}\in\{0,1\}^\omega\big\}$ of some (multigeometric) sequences $x=(x_n)_{n\in\omega}$.Comment: 10 page

Lineability of functions in C(K)C(K) with specified range

This paper is inspired by the paper of Leonetti, Russo and Somaglia [\textit{Dense lineability and spaceability in certain subsets of $\ell_\infty$.} Bull. London Math. Soc., 55: 2283--2303 (2023)] and the lineability problems raised therein. It concerns the properties of $\ell_\infty$ subsets defined by cluster points of sequences. Using the fact that the set of cluster points of a sequence $x$ depends only on its equivalence class in $\ell_\infty/c_0$ and that the quotient space $\ell_\infty/c_0$ is isometrically isomorphic to $C(\beta\mathbb{N}\setminus\mathbb{N})$, we are able to translate lineability problems from $\ell_\infty$ to $C(\beta\mathbb{N}\setminus\mathbb{N})$. We prove that for a compact space $K$ with properties similar to those of $\beta\mathbb{N}\setminus\mathbb{N}$, the sets of continuous functions $f$ in $C(K)$ with $\vert\operatorname{rng}(f)\vert=\omega$ and those $f$ with $\vert\operatorname{rng}(f)\vert=\mathfrak c$ contain, up to zero function, an isometric copy of $c_0(\kappa)$ for uncountable cardinal $\kappa$. Specializing those results to some closed subspaces $K$ of $\beta\mathbb{N}\setminus\mathbb{N}$ we are able to generalize known results to their ideal versions