Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

A set is called rational if it is well approximable by finite unions of arithmetic progressions, meaning that for every 0 > there exists a set , where , such that for all ; (b) is an averaging set of polynomial single recurrence; (c) is an averaging set of polynomial multiple recurrence. As an application, we show that if is rational and divisible, then for any set , where and Examples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form , then the following are equivalent: (a) is divisible, i.e. 0$]] is Euler's totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. Among other things, we show that if is a rational set with 0$]] with 0]]> and any polynomials , , which satisfy and for all , there exists 0 > such that the set STIX eqnarray > has positive lower density. Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences (sequences whose level-sets are rational). We prove that if is a finite alphabet, is rationally almost periodic, denotes the left-shift on and then is a generic point for an -invariant probability measure on such that the measure-preserving system is ergodic and has rational discrete spectrum.