Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

A set R subset of N is called rational if it is well approximable by finite unions of arithmetic progressions, meaning that for every epsilon > 0 there exists a set B = boolean OR(r)(i=1) a(i) N + b(i), where a(1),..., a(r), b(1),..., b(r) is an element of N, such that (d) over bar (R Delta B) := lim sup(N ->infinity) vertical bar(R Delta B) boolean AND (1,..., N}vertical bar/N < epsilon. 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 Phi(x) := {n is an element of N : phi(n)/n < x}, where x is an element of [0, 1] and phi 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 R subset of N is a rational set with (d) over bar (R) > 0, then the following are equivalent: (a) R is divisible, i.e. (d) over bar (R boolean AND uN) > 0 for all u is an element of N; (b) R is an averaging set of polynomial single recurrence; (c) R is an averaging set of polynomial multiple recurrence. As an application, we show that if R subset of N is rational and divisible, then for any set E subset of N with (d) over bar (E) > 0 and any polynomials p(i) is an element of Q[t], i = 1,..., l, which satisfy p(i)(Z) subset of Z and p(i) (0) = 0 for all i is an element of {1,..., l}, there exists beta > 0 such that the set {n is an element of R : (d) over bar (E boolean AND (E - p(1)(n)) boolean AND...boolean AND (E - p(l)(n))) > beta} 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 A is a finite alphabet, eta is an element of AN is rationally almost periodic, S denotes the left-shift on A(Z) and X := {y is an element of A(Z) : each word appearing in y appears in eta}, then eta is a generic point for an S-invariant probability measure nu on X such that the measure-preserving system (X, nu, S) is ergodic and has rational discrete spectrum.