A set R ⊂ N is called rational if it is well-approximable by finite unions of arithmetic progressions, meaning that for every ǫ > 0 there exists a set B = r i=1 a i N+b i , where a 1 , . . . , a r , b 1 , . . . , b r ∈ N, such thatExamples 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 Φ x := {n ∈ N :< x}, where x ∈ [0, 1] and ϕ 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 ⊂ N is a rational set with d(R) > 0, then the following are equivalent:(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 ⊂ N is rational and divisible, then for any set E ⊂ N with d(E) > 0 and any polynomials p i ∈ Q[t], i = 1, . . . , ℓ, which satisfy p i (Z) ⊂ Z and p i (0) = 0 for all i ∈ {1, . . . , ℓ}, there exists β > 0 such that the sethas 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, η ∈ A N is rationally almost periodic, S denotes the left-shift on A Z and X := {y ∈ A Z : each finite word appearing in y appears in η}, then η is a generic point for an S-invariant probability measure ν on X such that the measure preserving system (X, ν, S) is ergodic and has rational discrete spectrum.

In this paper we show that every set A ⊂ N with positive density contains B + C for some pair B, C of infinite subsets of N, settling a conjecture of Erdős. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.

We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if X = G/Γ is a nilmanifold, a 1 , . . . , a k ∈ G are commuting nilrotations, and f 1 , . . . , f k are functions of polynomial growth from a Hardy field then we show that• the distribution of the sequence aΓ is governed by its projection onto the maximal factor torus, which extends Leibman's Equidistribution Criterion form polynomials to a much wider range of functions; andΓ is always a finite union of sub-nilmanifolds, which extends some of the previous work of Leibman and Frantzikinakis on this topic.

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We investigate how spectral properties of a measure preserving system (X, B, µ, T ) are reflected in the multiple ergodic averages arising from that system. For certain sequences a : N → N we provide natural conditions on the spectrum σ(T ) such that for all f 1 , .In particular, our results apply to infinite arithmetic progressions a(n) = qn+r, Beatty sequences a(n) = ⌊θn+γ⌋, the sequence of squarefree numbers a(n) = q n , and the sequence of prime numbers a(n) = p n .We also obtain a new refinement of Szemerédi's theorem via Furstenberg's correspondence principle.