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

TL;DR

This paper generalizes the concept of $ ext{Lambda}(p)$-sets to $ ext{Lambda}( ext{Phi})$-sets within Orlicz spaces, establishing size estimates and constructing examples that distinguish different Orlicz functions, extending classical results.

Contribution

It introduces $ ext{Lambda}( ext{Phi})$-sets in Orlicz spaces, provides size estimates, and constructs examples differentiating these sets for various Orlicz functions.

Findings

Establishes size estimates for $ ext{Lambda}( ext{Phi})$-sets.

Constructs $ ext{Lambda}( ext{Phi}_1)$-sets not belonging to any $ ext{Lambda}( ext{Phi}_2)$-set under certain conditions.

Extends classical results on $ ext{Lambda}(p)$-sets to the more general $ ext{Lambda}( ext{Phi})$-sets.

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.