A variant of the $\Lambda (p)$ set problem in Orlicz spaces

Abstract

We introduce $\Lambda (\Phi )$-sets as generalizations of $\Lambda (p)$-sets. These sets are defined in terms of Orlicz norms. We consider $\Lambda (\Phi )$-sets when the Matuszewska-Orlicz index of $\Phi$ is larger than $2$. When $S$ is a $\Lambda (\Phi )$-set, we establish an estimate of the size of $S\cap [-N,N]$ where $N\in \mathbb{N}$. Next, we construct a $\Lambda ({\Phi }_1)$-set which is not a $\Lambda ({\Phi }_2)$-set for any ${\Phi }_2$ such that ${\sup }_{u\ge 1}{\Phi }_2(u)/{\Phi }_1(u)=\infty$ by using a probabilistic method. With an additional assumption about a subset $E$ of $\mathbb{Z}$, we can construct such a $\Lambda ({\Phi }_1)$-set contained in $E$. These statements extend known results on the structure of $\Lambda (p)$-sets to $\Lambda (\Phi )$-sets.

21 pages, corrected typos, updated references, removed the appendix, minor changes in the assumptions of theorem 1.4-1.6 and lemma 2.5 and the proof of lemma 4.3. Published in Math. Z