Abstract

Let (X, B, μ, T) be an ergodic measure-preserving system, let A ∈ B and let ϵ > 0. We study the largeness of sets of the form Ak+1 for various families of sequences. For and, we show that has positive density if, where satisfies or and denotes the th prime; or when is a certain Hardy field sequence. If is ergodic for some, then, for all, is syndetic if f(n) = qn + r. For, where are distinct integers, we show that S can be empty for K ≥ 4, and for k = 3, we found an interesting relation between the largeness of and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the fi are distinct polynomials.