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

Abstract

We introduce $ Λ(Φ) $-sets as generalizations of $ Λ(p) $-sets. These sets are defined in terms of Orlicz norms. We consider $Λ(Φ)$-sets when the Matuszewska-Orlicz index of $ Φ$ is larger than $ 2 $. When $S$ is a $Λ(Φ)$-set, we establish an estimate of the size of $ S \cap [-N,N] $ where $ N \in \mathbb{N} $. Next, we construct a $ Λ(Φ_1)$-set which is not a $ Λ(Φ_2)$-set for any $ Φ_2 $ such that $ \sup_{u \geq 1} Φ_2(u) / Φ_1(u) = \infty $ by using a probabilistic method. With an additional assumption about a subset $E$ of $\mathbb{Z}$, we can construct such a $Λ(Φ_1)$-set contained in $E$. These statements extend known results on the structure of $ Λ(p) $-sets to $Λ(Φ)$-sets.