Additive and multiplicative structures in completely I-large* sets

Abstract

It is known that for any IP⁎ set A and any sequence 〈xn〉n=1∞ in (N,+), there exists a sum subsystem of 〈xn〉n=1∞ whose finite sums and finite products are in A. Similar results have been established for some other combinatorial notions. In this article, we introduce a general notion called completely I-large⁎ sets, which can be IP⁎ sets, central⁎ sets or C⁎-sets when I is the corresponding ideal. And we establish a result for this notion to unify all known results. Furthermore, we apply this general result to another notion - quasi-central⁎ sets to obtain corresponding combinatorial results.