Multiplicatively large sets and ergodic Ramsey theory

Abstract

Multiplicatively large sets are defined in (ℕ, ·) by an analogy to sets of positive upper density in (ℕ, +). By utilizing various ergodic multiple recurrence theorems, we show that multiplicatively large sets have a rich combinatorial structure. In particular, it is proved that for any multiplicatively large setE ⊂ ℕ and anyk ∈ ℕ, there existsa,b,c,d,e,q ∈ ℕ such that {fx23-1}