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 \( \sup _{u \ge 1} \Phi _2(u) / \Phi _1(u) = \infty \) 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.