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 \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.