Abstract

Let (Xm B, mu, T) be an ergodic measure-preserving system, let A is an element of B and let epsilon > 0. We study the largeness of sets of the form S = { n is an element of N: mu(A boolean AND T-f(1)((n)) A boolean AND T-f2(n) A boolean AND center dot center dot center dot boolean AND T-f(k)((n)) A) > mu (A)(K+1) - epsilon} for various families {f(1); ..., f(k}) g of sequences f(i) (p(n)) N -> N. For k <= 3 and f(i) (n) = i f(n), we show that S has positive density if f(n) D q = p(n), where q is an element of Z[x ] satisfies q (1) or q(-1) = 0 and p(n) denotes the nth prime; or when f is a certain Hardy field sequence. f(i)(n) T-q is ergodic for some q is an element of N, then, for all r is an element of Z, S is syndetic if f (n) = qn + r. For f(i)(n) = a(i)n, where ai are distinct integers, we show that S can be empty for k >= 4, and, for k D 3, we found an interesting relation between the largeness of S 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.