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

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 $\sup_{u \geq 1} \Phi_2(u) / \Phi_1(u) = \infty$ 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.