Optimal lower bounds for multiple recurrence

Let be an ergodic measure-preserving system, let and let . We study the largeness of sets of the form 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 . For , where are distinct integers, we show that can be empty for , and, for , 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 are distinct polynomials.